objective Ques (315 results)
SSC CGL Mains 20241)
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SSC CGL Mains 20242)
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SSC CGL Mains 20243)
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SSC CGL Mains 20244)
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SSC CGL Mains 20245)
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6)If \(A = sin^2θ+cos^4θ\) for any value of θ, then the value of A is
7)If \(sin {πx\over 2}=x^2-2x+2\), then the value of x is
8)If \(cosπx=x^2-x+ {5 \over 4}\), the value of \(x\) will be
9)Which of the following is true for 0° < θ < 90° ?
11)The equation \(cos^2 θ= {(x+y)^2\over 4xy}\) is only possible when
12)The minimum value of sin2θ + cos2θ + sec2θ + cosec2θ + tan2θ + cot2θ
13)If 0° < θ < 90°, the value of sinθ + cosθ is
14)The minimum value of 2 sin2θ + 3 cos2θ is
15)sin(40° +θ)⋅cos(10° +θ)−cos(40° +θ)⋅sin(10° +θ) is equal to
16)If \(sin (30° + θ) = {3 \over \sqrt{12}}\) then cos2θ is
¾
17)The value of sin 50° – sin 70° + sin 10° is equal to
0
sin 50° – sin 70° + sin 10°
= sin(60° – 10°) – sin(60° + 10°) + sin 10°
Using the formulas
sin(A – B) = sin A cos B – cos A sin B
sin(A + B) = sin A cos B + cos A sin B, we get;
sin 50° – sin 70° + sin 10° = sin 60° cos 10° – cos 60° sin 10° – sin 60° + cos 10° – cos 60° sin 10° + sin 10°
= -2 cos 60° sin 10° + sin 10°
= -2 × (1/2) × sin 10° + sin 10°
= – sin 10° + sin 10°
= 0
18)If tan A = 1/2 and tan B = 1/3, then the value of A + B is
19)\(\rm \frac{cosA+cosB}{sinA-sinB}=\)
20)a.sinθ + b.cosθ = c, then find the value of a.cosθ - b.sinθ
21)sin 3 θ+cos3θ = 0, then the value of θ is
22)If sinθ + sin2θ + sin3θ = 1, then cos6θ - 4cos4θ + 8cos2θ is
23)If cosecθ - cotθ = \({1 \over a}\), then \( {a^2-1 \over a^2+1}\) is
24)\(\sqrt{sec^2 \theta+cosec^2\theta}\) is equal to
25)\(({{sinA + sinB }\over {cosA+cosB}})+({{cosA-cosB} \over{sinA-sinB}})\) is equal to:
0
26)If sin(A+B-C) = cos(A+C-B) = tan(B+C-A), then the value of angle A is
27)If cosec39° = x, the value of \({1 \over cosec^251°}+sin^239°+tan^251°-{1 \over sin^2 51°sec^2 39°} \) is
28)If A & B are complementary angles, then the value of sinA.cosB + cosA.sinB - tanA.tanB + sec2A - cot2B is
29)If x & y are acute angle, x+y <90° & sin(2x - 20°) = cos(2y + 20°), the value of sec(x + y) is
30)The value of \(cot{π\over 20}.cot{3π\over 20}.cot{5π\over 20}.cot{7π\over 20}.cot{9π\over 20}\) is
31)The value of cos20° + cos40° + cos60° + ...... + cos160° + cos180°
-1
32)The value of cos24° + cos55° + cos125° + cos204° + cos 300° is
0.5
33)The value of (1 + sec20° + cot70°)(1 - cosec20° + tan70°) is:
34)the value of \(sin^21°+sin^25°+sin^29°+............... +sin^289°\):
\(11{1\over2}\)
35)If A, B, and C are the angels of a △ABC then the following is equal to :
sin\(({B+C \over2})\)
36)If \(sin21°={x\over y}\), then \(sec21°-sin69°\) is equal to
\(x^2\over{y}\sqrt{y^2-x^2}\)
37)If \(\alpha+\beta=90°\) , then the value of \({(1-sin^2\alpha)(1-cos^2\alpha)*(1+cot^2\beta)(1+tan^2 \beta)}\) is.
38)If \({ sec^270° -cot^220°\over2{(cosec^259°-tan^231°)} }= {2 \over m}\)then m is equal to:
39)Evaluate: 3 cos 80° cosec 10° + 2 cos 59° cosec 31°
40)If sin\({(60°-\theta)}\)=cos\({(\psi-30°)}\), then the value of tan \({(\psi-\theta)}\) is {assume that \(\theta\) and \(\psi\) are both positive acute angels with \(\theta<60°\) and \(\psi>30°)\).
\( \sqrt{3} \)
41)If \(2sin^2 \theta-3sin\theta + 1 = 0;\) θ being positive acute angle(s), then the value of θ is/are
30°, 90°
42)sin (A+B) = 1 and cos (A-B) = √3/2, where A & B are positive acute angles with A ≥ , then A and B are:
A = 60°, B = 30°
43)If \(\alpha+\theta= {7 \pi \over 12}\) and tanθ = √3, then the value of tanα is
1
44)If \(7sin^2θ+3cos^2 θ = 4\) (0° ≤ θ ≤ 90°), then value of θ is
π/6
45)If \(sec^2θ+tan^2 = 7\), then the value of θ where 0° ≤ θ ≤ 90° is
46)If secθ + tanθ = √3 (0° ≤ θ ≤ 90°) then the value of tan3θ = ?
47)If tan (θ1 + θ2) = √3 and sec (θ1 - θ2) = 2/√3 , then the value of sin2θ1 + tan3θ2 is equal to (assume 0°< θ1 - θ2 < θ1 + θ2 < 90°
48)The value of cos1°. cos2°.cos3°..........cos179° is
49)The value of x in the equation \(tan^2{π\over 4}-cos^2{π\over 3} = x sin{π\over 4}.cos{π\over 4}.tan{π\over 3}\) is:-
50)If \({xcosec^230°. sec^245° \over 8 cos^245°.sin^260° } = tan^260°- tan^230°,\) then the value of x is:-
1
51)Find the value of \(cot30°+cot75°+cot45°-sin90°+sin45° \over sin60°+cos30°+ tan15°+cos45°\)
52)If \(sin \theta = {a \over b}\), then the value of secθ - cosθ is where 0° < θ < 90°
\(a^2 \over {b \sqrt{b^2-a^2}}\)
53)If \(cos\theta = {p \over \sqrt {p^2+q^2}}\), then the value of tanθ is
\(q \over p\)
54)If \(tan^2θ = 1-e^2\) , then the value of \(secθ + tan^3 θ.cosecθ\) is:
55)If \(\sec^2θ = 3\), 0° < θ < 90°, then the value of \(tan^2θ - cosec^2θ \over tan^2θ + cosec^2θ\) is :
1/7
56)If \(cot\theta = {9 \over 40} \) and \(\pi < θ < {3\pi \over 2} \), then the value of cosecθ is
-41/40
57)If \(sec \theta= {113 \over 112}\) and θ lies in the 4th Quadrant, then the value of sinθ is
58)If \(tan\theta = {a \over b}\), then \({a.sin\theta + b.cos\theta \over {a.sin\theta - b.cos\theta}}\) is:
59)If \(sec\theta = x + {1\over 4x}\) (0° < θ < 90°), then secθ + tanθ is equal to
SSC CGL 202260)If sin A = 4/5, then what is the value of sin2 A?
16/25
SSC CGL 202261)If \(cos θ + cos^2θ =1\), find the value of \(\sqrt{\sin^4θ + \cos^2θ}\)
\(\sqrt{2}cos θ \)
SSC CGL 202262)If \(sec A = {5\over4}\) , then the value of \(\rm \frac{\tan A}{1 + \tan^2A} - \frac{\sin A}{\sec A}\)is:
0
SSC CGL 202263)If \(2 sinθ + 2 sin^2θ=2\), then the value of \(2 cos^4θ + 2 cos^2θ\) is:
2
SSC CGL 202264)Find the value of s\(ecθ - tanθ, if secθ + tanθ = \sqrt5\) .
\(5\frac{\sqrt5}{5}\)
SSC CGL 202265)If tan3θ⋅tan7θ = 1, where 7θ is an acute angle, then find the value of cot15θ.
-1
SSC CGL 202266)\(\frac{{1 + \sin \theta }}{{\cos \theta }}\) is equal to which of the following (where \(\theta \ne \frac{\pi }{2}\))?
\(\frac{{\cos \theta }}{{1 - \sin \theta }}\)
SSC CGL 202267)If sin2 θ − 3 sin θ + 2 = 0, then find the value of θ (0° ≤ θ ≤ 90°).
90°
SSC CGL 202268)Find the value of tan 27° tan 34° + tan 34° tan 29° + tan 29° tan 27°.
SSC CGL 202269)Find the value of cos 2A cos 2B + sin2(A - B) - sin2(A + B)
cos (2A + 2B)
SSC CGL 202270)If cosec θ + cot θ = p, then the value of \({{p^2 \ - \ 1} \over p^2 \ + \ 1}\) is:
cos θ
SSC CGL 202271)If θ is an acute angle and tan θ + cot θ = 2, then the value of \(tan^{200} θ + cot^{200} θ\) is:
SSC CGL 202272)If sec2A + tan2A = 3, then what is the value of cot A?
1
SSC CGL 202273)\(\rm {\cos A \over {1 - \tan A}} + {\sin A \over {1 - \cot A}}\) = ________.
sin A + cos A
SSC CGL 202274)What is the value of cosec 15° sec 15°?
4
SSC CGL 202275)If a = 45° and b = 15°, what is the value of \({\cos (a - b ) - \cos (a + b)} \over {\cos(a - b) + \cos(a + b)}\) ?
2 - √3
SSC CGL 202276)If \( tan^2\theta + tan^4\theta = 1 \), then:
SSC CGL 202277)If \({tan40^0}={\alpha}\) , then find\(\frac{tan320^0 - tan 310^0}{1 + tan320^0.tan 310^0}\) .
\(\frac{1 - \alpha^2}{2\alpha}\)
SSC CGL 202278)If \(sec \theta + \frac{1}{cos \theta}=2\), find the value of \(sec^{55} \theta + \frac{1}{sec ^{55} \theta}\).
SSC CGL 202279)If A is an acute angle, the simplified form of
\(\rm \frac{{\cos (\pi - A).\cot \left( {\frac{\pi }{2} + A} \right)\cos ( - A)}}{{\tan (\pi + A)\tan \left( {\frac{{3\pi }}{2} + A} \right)\sin (2\pi - A)}}\) is :
Cos A
SSC CGL 202280)If tan (α + β) = a, tan (α - β) = b, then the value of tan 2α is :
\(\rm \frac{a+b}{1-ab}\)
SSC CGL 202281)If tan2 θ = 1 - a2, then the value of sec θ + tan3 θ cosec θ is:
\((2 - a^2)^{{3} \over 2}\)
SSC CGL 202282)If \(Cos A = {{9} \over 41} \), find cot A.
\({{9} \over 40} \)
SSC CGL 202283)\(16 \cos^3 {{\pi } \over 6} - 12 \cos {{\pi } \over 6} =\) _______________.
0
SSC CGL 202284)If tan θ + sec θ= 7, being acute, then the value of 5 sin is:
\({{24} \over 5}\)
SSC CGL 202285)What is the value of tan 240°?
\(\sqrt3\)
SSC CGL 202286)The value of \({{2 \tan 60°} \over 1+ \tan^260°}\) is:
sin 60°
SSC CGL 202287)What is sin α - sin β?
\(2 \cos \frac{α+β}{2} \sin \frac{α-β}{2}\)
SSC CGL 202288)The value of (sin 30° cos 60°- cos 30° sin60°) is equal to:
-sin 30°
SSC CGL 202289)What will be the value of \(\frac{\sin 30^{\circ} \sin 40^{\circ} \sin 50^{\circ} \sin 60^{\circ}}{\cos 30^{\circ} \cos 40^{\circ} \cos 50^{\circ} \cos 60^{\circ}} \)?
1
SSC CGL 202290)The value of tan2θ + cot2θ - sec2θ cosec2θ is:
-2
SSC CGL 202291)If cosec A = sec B, where A and B are acute angles, then what is the value of (A + B)?
90°
SSC CGL 202292)If \(\frac{\sin ^{2} ∅-3 \sin ∅+2}{\cos ^{2} ∅}=1 \), where 0° < ∅ < 90°, then what is the value of (cos 2∅ - sin 3∅ + cosec 2∅)?
\(\frac{-3+4 \sqrt{3}}{6}\)
SSC CGL 202293)If sec2 α + 4 cos2α = 4 and 0° ≤ α ≤ 90°, then find the value of α.
SSC CGL 202294)If \(tan A = \frac{{2.4}}{{0.7}}\) ,what is the value of (50 cos A + 24 cot A)?
21
SSC CGL 202295)If (2 cos A + 1) (2 cos A - 1) = 0, 0° < A ≤ 90°, then find the value of A.
60°
SSC CGL 202296)If A = 30°, what is the value of: \(\frac{\left[8 \sin \mathrm{A}+11 \operatorname{cosec} \mathrm{A}-\cot ^{2} \mathrm{A}\right]}{10 \cos 2 \mathrm{A}}\) ?
\(4 \frac{3}{5}\)
SSC CGL 202297)The value of \(\frac{5 \cos ^{2} 62^{\circ}+5 \cos ^{2} 28^{\circ}-21}{7 \sin ^{2} 35^{\circ}+7 \sin ^{2} 55^{\circ}+1}\) is:
-2
SSC CGL 202298)The value of \(\left(\frac{1-\cot θ}{1-\tan θ}\right)^{2} + 1\), if 0° < θ < 90°, is equal to:
cosec2θ
SSC CGL 202299)If tan2 A - 6 tan A + 9 = 0, 0 < A < 90°, what is the value of \(6 cot A + 8 \sqrt{10} cos A\)?
10
SSC CGL 2022100)If tan B =5/3, then what is the value of \( cosec B +sin B \over cos B - secB\) ?
\(- {177\over125}\)
SSC CGL 2022101)If 6tanA (tanA + 1) = 5 - tanA, Given that \( 0 < A< \frac{\pi}{2}\) what is the value of (sinA + cosA)?
\(\frac{3}{{\sqrt 5 }}\)
SSC CGL 2022102)Simplify the following expression:
cosec4 A(1 - cos4 A) - 2 cot2 A - 1
0
SSC CGL 2022103)Simplify the following expression:
\( \frac{{\cos A}}{{1 - \tan A}} + \frac{{\sin A}}{{1 - \cot A}} - \sin A\)
cos A
SSC CGL 2022104)If cot2 α + tan2 α = 2, 0° ≤ α ≤ 90°, then find the value of α
45°
SSC CGL 2022105)If cos 53° = x / y , then sec 53° + cot 37° is equal to:
\(\frac{{y + \sqrt {{y^2} - {x^2}} }}{x}\)
SSC CGL 2022106)If cos B = 5 / 7 , what is the value of cosec B + cot B? Given that 0 < B < π/2?
\(\sqrt 6\)
SSC CGL 2022107)If 3 sin2 θ + 4cos θ - 4 = 0, 0° < θ < 90, then the value of (cosec2 θ + cot2 θ) is
5/4
SSC CGL 2022108)If A = 60°, what is the value of:
\( \frac{{{\rm{10 \ sin }}\frac{{\rm{A}}}{{\rm{2}}}{\rm{ \ + \ 8cosA}}}}{{{\rm{7\ sin }}\frac{{{\rm{3A}}}}{{\rm{2}}}{\rm{ \ - \ 12cosA}}}}={\rm{?}}\)
9
SSC CGL 2022109)If 2 sin2 θ + 3 cos θ = 3, 0∘ < θ < 90∘ , then the value of ( sec2 θ + cot2 θ) is
\(4\frac{1}{3} \)
SSC CGL 2022110)If A = 10∘ , what is the value of: \(\frac{{12\sin 3A + 5\cos (5A - {5^ \circ })}}{{9\sin \frac{{9A}}{2} - 4\cos (5A + {{10}^ \circ })}}\)
\(\frac{{6\sqrt 2 + 5}}{{(9 - 2\sqrt 2 )}} \)
SSC CGL 2022111)The value of 2 - \(\sqrt {\frac{{\cot \theta + \cos \theta }}{{\cot \theta - \cos \theta }}}\), when 0° < θ < 90° is equal to:
2 – sec θ – tan θ
SSC CGL 2022112)The expression (cos6 θ + sin6 θ-1) (tan2θ + cot2θ + 2) +1 is equal to:
-2
SSC CGL 2022113)If sec2θ + tan2θ = 3\( \frac{1}{2}\), 0° < θ < 90° then (cosθ + sinθ) is equal to
\(\frac{2+\sqrt{5}}{3}\)
SSC CGL 2022114)If A = 60°, what is the value of:
\(\frac{\left[8 \cos \mathrm{A}+7 \sec \mathrm{A}-\tan ^{2} \mathrm{~A}\right]}{10 \sin \frac{A}{2}}\)
3
SSC CGL 2022115)\(\begin{aligned} (\sec \emptyset-\tan \emptyset)^{2}(1+\sin \emptyset)^{2} \div \cos ^{2} \emptyset=? \\ \end{aligned}\)
1
SSC CGL 2022116)If sin2θ - cos2θ - 3sinθ + 2 = 0, 0° < θ < 90°, then what is the value of 1 + secθ + tanθ?
SSC CGL 2022117)The expression (cos6 θ + sin6 θ – 1)(tan2 θ + cot2 θ + 2) + 3 is equal to:
0
SSC CGL 2022118)If 5 sin θ – 4 cos θ = 0, 0° < θ < 90°, then the value of \(\frac{5sin\theta-cos\theta}{5sin\theta+3cos\theta}\) is:
\(\frac{3}{7}\)
SSC CGL 2022119)If 5 sin θ - 4 cos θ = 0, 0° < θ < 90°, then the value of \(\frac{{{\rm{5sin}}\,{\rm{\theta }}\,{\rm{ + }}\,{\rm{2 cos}}\,{\rm{\theta }}}}{{{\rm{5sin}}\,{\rm{\theta }}\,{\rm{ + }}\,{\rm{3 cos}}\,{\rm{\theta }}}}\) is:
\(\frac{6}{7}\)
SSC CGL 2022120)Find the value of the following expression:
\( \frac{{{{\tan }^3}45^\circ + 4{{\cos }^3}60^\circ }}{{2{\rm{cose}}{{\rm{c}}^2}45^\circ - 3{{\sec }^2}30^\circ + \sin 30^\circ }}\)
3
SSC CGL 2022121)If 2k sin 30° cos 30° cot 60° = \(\frac{{{{\cot }^2}30^\circ \sec 60^\circ \tan 45^\circ }}{{{\rm{cose}}{{\rm{c}}^{\rm{2}}}{\rm{45^\circ cosec 30}}^\circ }}\) , then find the value of k.
3
SSC CGL 2022122)tan2 A + 5 sec A = 13, where 0 < A < 90°. Solve for A (in degrees).
60
SSC CGL 2022123)The value of \( \left({ {1 - \cotθ} \over 1 - \tanθ}\right)^2 - 1\) when 0° < θ < 90°, is equal to:
cot2 θ - 1
SSC CGL 2022124)The value of \(1 + \sqrt { {\cotθ + \cosθ} \over \cotθ - \cosθ}\) , if 0° < θ < 90°, is equal to:
SSC CGL 2022125)If cot B = \(\frac{12}{5}\) , what is the value of sec B?
\(\frac{13}{12}\)
SSC CGL 2022126)If 3sec2θ + tanθ - 7 = 0, 0° < θ < 90°, then what is the value of \((\frac{2 sinθ +3 cosθ} {cosecθ +secθ})\) ?
\(\frac{5}{4}\)
SSC CGL 2022127)If sec A = 17/8, given that A < 90°, what is the value of the following? \(34 sin A+15 cot A \over 68 Cos A-16 tan A\)
19
SSC CGL 2022128)If tan2A + 2tanA - 63 = 0 Given that 0° < A < 90° what is the value of (2sinA + 5cosA)?
\(19 \over \sqrt{50}\)
SSC CGL 2022129)If cos (A - B) = √3/2 and secA = 2, 0º ≤ A ≤ 90º, 0º ≤ B ≤ 90º, then what is the measure of B?
30º
SSC CGL 2022130)What is the value of: 8 √3 sin 30º tan 60º - 3 cos 0º + 3 sin2 45º + 2 cos2 30º?
12
SSC CGL 2022131)The value of: \(\frac{{\sin 23^\circ \cos 67^\circ + \sec52^\circ \sin38^\circ + \cos 23^\circ \sin 67^\circ + \rm cosec52^\circ \cos 38^\circ }}{{\rm cose{c^2}20^\circ - {{\tan }^2}70^\circ }}\) is
3
SSC CGL 2022132)If 4sin2 θ = 3(1+ cos θ), 0° < θ < 90°, then what is the value of (2tan θ + 4sin θ - sec θ)?
3√15 - 4
SSC CPO 2020133)Solve for θ: \({\cos ^2}\theta - {\sin ^2}\theta = \frac{1}{2} \), 0 < θ < 90°.
30°
SSC CPO 2020134)If cotθ= 1/√ 3 and 0° < θ° < 90°, then the value of \( \frac{{2 - {{\sin }^2}\theta }}{{1 - {{\cos }^2}\theta }} + (cose{c^2}\theta - {\sec} \theta ) \) is:
1
SSC CPO 2020135)If 4(cosec2 57 - tan2 33) - cos 90 + y × tan2 66 × tan2 24 = y/2, then the value of y is:
-8
SSC CPO 2020136)If 4 - 2sin2θ - 5cosθ = 0, 0°< θ < 90°, then the value of cosθ - tanθ is:
\(\frac{{1 - 2\surd 3}}{2}\)
SSC CPO 2020137)The value of \(\frac{{{{\sin }^2}{}52^\circ + 2 + {{\sin }^2}{{ }}38^\circ }}{{4{}{{\cos }^2}43^\circ - 5 + 4{{\cos }^2}{{}}47^\circ }}\) is:
- 3
SSC CPO 2020138)If 4θ is an acute angle, and cot 4θ = tan (θ - 5°) , then what is the value of θ?
19°
SSC CPO 2020139)If sin3x = cos(3x – 45°), 0° < 3x < 90°, then x is equal to:
22.5°
SSC CPO 2020140)The value of \(\frac{{{{\sin }^2}30^\circ + {{\cos }^2}60^\circ + sec45^\circ .sin45^\circ }}{{sec60^\circ + cosec30^\circ }} \) is:
\(\frac{3}{8}\)
SSC CPO 2020141)If sec3x = cosec(3x - 45°), where 3x is an acute angle, then x is equal to:
22.5°
SSC CPO 2020142)The value of \(\frac {\sin^2 30^\circ + \cos^2 60^\circ - \sec 35^\circ.\sin 55^\circ}{\sec 60^\circ + \rm cosec\;30^\circ} \) is equal to:
\(-\frac 1 8\)
SSC CPO 2020143)If 4(cosec2 57° - tan2 33°) - cos 90° - y tan2 66° tan2 24° =\(\dfrac{y}{2}\) , the value of y is:
\(\dfrac{8}{3}\)
SSC CPO 2020144)If 4 – 2sin2 θ – 5cos θ = 0, 0° < θ < 90°, then the value of cos θ + tan θ is :
\(\dfrac{1+2\sqrt{3}}{2}\)
SSC CHSL 2021145)If \(tan A + sec A ={3\over2}\) and A is an acute angle, then the value of \( \frac{10\cot A+13\cos A}{12 \tan A + 5 \ cosec \ A} \) is:
2
SSC CHSL 2021146)If a triangle ABC is right-angled at A, then what is the value of \(\rm \sin\frac{B+C}{2}\cos\frac{B+C}{2}\)?
\(\frac{1}{2}\)
SSC CHSL 2021147)If sec θ - cosec θ = 0 and θ is an acute angle, then what is the value of sec2 θ + cosec2 θ ?
4
SSC CHSL 2021148)If cosec A = 10, then what is the value of 20 sin A + 9√11 sec A?
Given that A is an acute angle.
32
SSC CHSL 2021149)If sin2x = 3 cos2x and 0° ≤ x ≤ 90°, then what is the value of \(\frac{x}{2} \)?
30°
SSC CHSL 2021150)If 5 tan A = 12, then what is the value of \(\frac{13 \ sin A \ + \ 20 \ tan A}{15 \ tan A \ - \ 13 \ cos A}\), where A is an acute angle?
\(1\frac{29}{31}\)
SSC CHSL 2021151)The value of cosec (58° + θ) - sec (32° - θ) + sin 15° sin 35° sec 55° sin 30° sec 75° is:
\(\frac{1}{2}\)
SSC CHSL 2021152)If tan θ + cot θ = 3, then what will be the value of tan2 θ + cot2 θ ?
7
SSC CHSL 2021153)If tan 3θ = sin 45°. cos 45° + cos 60° and 3 θ is an acute angle, then what will be the value of sin 4θ?
\(\frac{\sqrt3}{2}\)
SSC CHSL 2021154)The value of \(\frac{cos^2 89^\circ \ + \ cos^2 1^\circ}{cos 30^\circ \ sin90^\circ \ - \ sin30^\circ cos90^\circ}\) is:
\(\frac{2}{\sqrt{3}}\)
SSC CHSL 2021155)If 6tan2 α - 2 = 0, (0 < α < 90°), then the value of α is:
30°
SSC CHSL 2021156)If \(sin α + csc α = tan{\pi\over 3}\), then the value of (sin3α + csc3α) is equal to:
SSC CHSL 2021157)If cot A + cosec A = 2 and A is an acute angle, then the value of \(\frac{9\tan A+16\ cosec\ A}{5\sin A+3\tan A} \) is:
4
SSC CHSL 2021158)Solve for θ : 3 cosec θ + 4 sin θ - 4√3 = 0, where θ is an acute angle.
60°
SSC CHSL 2021159)If cot2 θ + cot4 θ = 2, then the value of 2 sin4 θ + sin2 θ is:
SSC CHSL 2021160)The value of \([\frac{\sin^227^{\circ}+\sin^263^{\circ}}{\cos^224^{\circ}+\cos^266^{\circ}}\)\(-\sin^269^{\circ}-\)\(\cos69^{\circ}\sin21^{\circ}]\) is:
0
SSC CHSL 2021161)The value of \(\frac{\sec \theta \ cosec \theta}{2+\tan^2 \theta+\cot^2 \theta}\) is equal to:
sinθ cosθ
SSC CHSL 2021162)If 3 sin2 A + 4 cos2 A - 3 = 0, then the value of cot A (where 0 ≤ A ≤ 90°) is:
0
SSC CHSL 2021163)The value of cot246° - sec244° + (sin21° + sin 23° + sin25° + ....+ sin289°) is:
21.5
SSC CHSL 2021164)If 3 cos θ = 2 sin2 θ, \(0^0<\theta<90^0\), then what is the value of (tan2 θ + sec2 θ - cosec2 θ)?
\(\frac{17}{3}\)
SSC CHSL 2021165)If (1 + cot2θ) + (1 +(cot2θ)-1) is equal to k, then √k = ?
SSC CHSL 2021166)The value of \(\sqrt{\cos 60^{\circ} \cos 30^{\circ}- \sin60^{\circ} \sin 30^{\circ}}\) is:
0
SSC CHSL 2021167)If \({5cot\theta+\sqrt3cosec\theta\over 2\sqrt3coesec\theta+3 cot\theta}=1\), \(0^0<\theta<90^0\), then the value of \(\frac{\frac{7}{2} \cot^2 \theta- \frac{3}{4} \:cosec^2 \theta}{4 \sin^2 \theta+\frac{3}{2} \tan^2 \theta}\) will be:
SSC CHSL 2021168)If sin2 x - 3cos2 x = 0, then the value of x (0 < x < 90°) is:
60
SSC CHSL 2021169)For θ : 0° < θ < 90°
3 sec θ + 4 cos θ = 4√3, find the value of (1 - sin θ + cos θ).
\(\frac{1+\sqrt3}{2}\)
SSC CHSL 2021170)Evaluate the following expression.
\( \frac{3(\cot^2 46^{\circ}-\sec^2 44^{\circ})}{2(\sin^2 28^{\circ}+\sin^2 62^{\circ})}+\)\(\frac{2\cos^2 60^{\circ}\tan^2 33^{\circ}\tan^2 57^{\circ}}{\sec^2(90^0-\theta)-\cot^2 \theta}\)
-1
SSC CHSL 2021171)If \(tan\theta ={4\over3}\), then the value of \(\frac{9\sin \theta+12\cos \theta}{27\cos \theta-20\sin \theta} \) will be equal to:
72
SSC CHSL 2021172)If \( sin B =\frac{9}{41}\) , then what is the value of cot B, where 0° < B < 90°?
\(\frac{40}{9}\)
SSC CHSL 2021173)The value of \( \frac{cos 8^\circ \ cos 24^\circ \ cos 60^\circ \cos 66^\circ \ cos 82^\circ}{sin 82^\circ sin 66^\circ \ sin 60^\circ \ sin 8^\circ \ sin 24^\circ}\) is:
\(\frac{1}{\sqrt{3}}\)
SSC CHSL 2021174)For what value of θ (in degrees) is the following equation true?
\( sin 3θ cos θ - cos 3θ sin θ ={1\over2}\), \(0<\theta<{\pi\over2}\)
15
SSC CHSL 2021175)In a right-angled triangle ABC right angled at C, sin A = sin B. What is the value of cos A?
\(\frac{1}{\sqrt2}\)
SSC CHSL 2021176)If \({cosec^2\theta\over cosec^2\theta-cot^2\theta}={13\over4}\), \(0^0<\theta<90^0\), then the value of \(\frac{52 \cos^2 \theta-9 \tan^2 \theta}{18 \sec^2 \theta + 8 \cot^2 \theta}\) will be:
\(\frac{8}{11}\)
SSC CHSL 2021177)If 5 cos θ = 4 sin θ, 0° ≤ θ ≤ 90°, then what will be the value of sec θ?
\(\frac{\sqrt{41}}{4}\)
SSC CHSL 2021178)If \(\frac{\cot \theta + \cos \theta}{\cot \theta - \cos \theta}\) \(=\frac{k+1}{1-k}\), \(k \ne 1\), then k is equal to:
sinθ
SSC CHSL 2021179)If cos2θ - sin2θ - 3 cos θ + 2 = 0, \(0^0<\theta<90^0\), then what will be the value of sec θ - cos θ?
\(\frac{3}{2}\)
SSC CHSL 2021180)If √13 sin θ = 2, then the value of \(\frac{3\tan \theta+\sqrt{13} \sin\theta}{\sqrt{13} \cos \theta - 3\tan \theta}\) is:
4
SSC CHSL 2021181)What is the value of \(\frac{\cos^2 20^{\circ}+\cos^2 70^{\circ}}{\sin^2 90^{\circ}}\)\(-\tan^2 45^{\circ}\) ?
0
SSC CHSL 2021182)If 3 cot A = 4 tan A and A is an acute angle, then what will be the value of sec A ?
\(\frac{\sqrt7}{2}\)
SSC CHSL 2021183)If \({sin^2\theta\over tan^2\theta-sin^2\theta}=5\), then the value of \(\frac{24\cos^2\theta-15\sec^2\theta}{6\ \rm cosec^2 \theta-7\cot^2\theta}\) is:
SSC CHSL 2021184)If sinθ + cosecθ = 7, then what is the value of sin3θ + cosec3θ?
322
SSC CHSL 2021185)In a triangle ABC, right-angled at C, if \(sec A={13\over5}\), then find the value of \(\frac{1+ \sin A}{\cos B}\).
SSC CHSL 2021186)Evaluate the following expression.
\(\frac{\tan^2 60^{\circ}+ cosec\: 30^{\circ}\sin 90^{\circ}+3 \sec^230^{\circ}}{4 \sin^2 45^{\circ}+ \sec^2 60^{\circ} - \cot^2 30^{\circ}-5 \cos^290^{\circ}}\)
3
SSC CHSL 2021187)If cosec2 θ (cos θ - 1)(1 + cos θ) = k, then what is the value of k?
-1
SSC CHSL 2021188)Find the value of θ, if sec2 θ + (1 - √3) tan θ - (1 + √3) = 0, where θ is an acute angle.
SSC CHSL 2021189)If \(cos\theta={7\over3\sqrt6}\) and θ is an acute angle, then the value of \(27\sin^2 \theta-\frac{3}{2}\) is:
1°
SSC CHSL 2021190)For A = 30°, find the value of: \(\frac{-3\sin^22A+2\sec^2A-\tan\frac{3A}{2}}{\frac{1}{3}\sin3A}\)
\(-\frac{7}{4}\)
SSC CHSL 2021191)The value of \(\frac{3 \tan^2 60^{\circ}+ \sec^2 30^{\circ}- \sin^2 45^{\circ}}{(\cos 15^{\circ}+ \sin 75^{\circ})(\sec15^{\circ}+ cosec 75^{\circ})}\) is:
\(\frac{59}{24}\)
SSC CHSL 2021192)If 7 sin2 θ + 3 cos2 θ = 4, 0° < θ < 90°, then the value of θ will be:
30°
SSC CHSL 2021193)If \( \cos \theta = \frac{P^2-1}{P^2+1}\), \(0^{\circ}<\theta<90^{\circ}\), then cosec θ is equal to:
\(\frac{1+P^2}{2P}\)
SSC CHSL 2021194)If \(cos\theta ={\sqrt3\over2}\), then the value of \(\frac{2 - \sin^2 θ}{1 - \cot^2 θ}\) + (sec2θ + cosecθ) is:
\(\frac{59}{24}\)
SSC CHSL 2021195)The value of \(\frac{\tan 50^\circ + \sec 50^\circ}{\cot 40^\circ + \text{cosec} \ 40^\circ}\)\( + cos^265° \)\(+ sin 65° cos 25°\)\( + tan 30°\) is:
\(\frac{6+\sqrt{3}}{3}\)
SSC CHSL 2021196)If tan x = cot (48° + 2x), and 0° < x < 90°, then what is the value of x?
14°
SSC CHSL 2021197)If 2cos2θ = 3 (1 – sinθ), 0° < θ < 90°, then what is the value of (tan2θ + cosec3θ - sec2θ)
\(\sqrt 3 - 1\)
SSC CHSL 2021198)In Δ ABC, ∠A = 90°, AB = 20 cm and BC = 29 cm. What is the value of (sinB – cotC)?
\(- \frac{189}{{580}}\)
SSC CHSL 2021199)Solve the following equation and find the value of θ.
3cot θ + tan θ - 2√3 = 0, 0 < θ < 90°
60°
SSC CHSL 2021200)If \(cosecθ = \frac{41}{9}\) and θ is an acute angle, then the value of 5 tan θ will be:
\(\frac{9}{8}\)
SSC CHSL 2021201)If (sin A - cos A) = 0, then what is the value of cot A?
1
SSC CHSL 2021202)If \(cosθ = \frac{4x}{1 + 4x^2}\), then what is the value of sin θ ?
\(\frac{1 - 4x^2}{1+4x^2}\)
SSC CHSL 2021203)If \( \rm \frac{\sin θ+\cos θ}{\sin θ-\cos θ}=3\), then the value of sin4 θ - cos4 θ is equal to:
\(\frac{3}{5}\)
SSC CHSL 2021204)If \(\rm \sec\left(90^{\circ}-\frac{3θ}{2}\right)=\sqrt2\) , 0° < θ < 90°, then the value of 2sinθ + 4cos2θ will be:
3
SSC CHSL 2021205)Solve the following equation.
2√3 sin2 θ + cos θ – √3 = 0 where θ is an acute angle.
30°
SSC CHSL 2021206)The value of \(\frac{3cos^227^\circ-5 + 3cos^263^\circ }{tan^232^\circ + 4 - cosec^2 58^\circ} \)\(+ sin35°cos55° \)\(+ cos35°sin55°\) is:
\(\frac{1}{3}\)
SSC CHSL 2021207)If \(sin\theta={2\sqrt{ab}\over a+b}\), a > b > 0, then the value of \( \frac{cos\theta + 1}{cos \theta - 1}\) will be:
\(-\frac{a}{b}\)
SSC CHSL 2021208)If 3cos2θ - 4sinθ + 1 , 0° < θ < 90°, then the value of 3cos2θ + 5tan2θ will be:
\(5\frac{2}{3}\)
SSC CHSL 2021209)\(\frac{(1+\cos \theta)(cosec \theta-\cot \theta)\sec \theta}{\sin \theta(1-\sin\theta)(\sec \theta+\tan \theta)}=?\)
sec2 θ
SSC CHSL 2021210)If \(\sin θ=\frac{11}{15} \), then the value of (sec θ - tan θ) is:
\(\frac{\sqrt{26}}{13}\)
SSC CHSL 2021211)If sin θ( 2 sin θ + 3) = 2, 0° < θ < 90°, then what is the value of (sec2 θ + cot2 θ - cos2 θ)?
\(\frac{43}{12}\)
SSC CHSL 2021212)If 5k = tanθ and \({5\over k}= sec\theta\), then what is the value of \(10\left(k^2-\frac{1}{k^2}\right) \) ?
\(-\frac{2}{5}\)
SSC CHSL 2021213)Simplify the following expression.
cos2 30° + cos2 40°+ cos2 50° + cos2 60°
2
SSC CHSL 2021214)If 21 tan θ = 20, then(1 + sin θ - cos θ) : (1 - sin θ + cos θ) is equal to:
14 ∶ 15
SSC CHSL 2021215)If 3(sec2θ + tan2θ) = 5, 0° < θ < 90°, then the value of cosec θ is:
2
SSC CHSL 2021216)If \((cos^2\theta-1)(2sec^2\theta)+\)\(sec^2\theta+2tan^2\theta=2\), 0° < θ < 90° , then the value of \(\dfrac{(sec\theta + sin\theta)}{(cosec \theta - cos \theta )}\)will be:
3
SSC CHSL 2021217)If \(sin\theta = \frac{12}{13}\), then \(\frac{sin^2\theta - cos^2 \theta }{2cos\theta sin \theta } \times cot^2\theta=?\)
\(\dfrac{595}{3456}\)
SSC CHSL 2021218)If A = 60°, then what is the value of (4cos3 A - 3cos A)?
-1
SSC CHSL 2021219)What is the value of \(\frac{ \sin 33^\circ \cos 57^\circ + \sec 62^\circ \sin 28^\circ + \cos 33^\circ \sin 57^\circ + \rm cosec 62^\circ \cos 28^\circ}{\tan 15^\circ \tan 35^\circ \tan 60^\circ \tan 55^\circ \tan 75^\circ}\) ?
√3
SSC CHSL 2021220)If tan2x - 3tanx + 2 = 0 and (0° < x < 90°), then the value of x is :
45°
SSC CHSL 2021221)If Y = tan35°, then the value of (2tan55° + cot55°) is :
\(\frac{2 + Y^2}{Y}\)
SSC CHSL 2021222)If tanθ + cotθ = 4, then the ratio of 3\((tan^2\theta+cot^2\theta)\) to (2cosec2θ sec2θ - 4) will be:
3 : 2
SSC CHSL 2021223)What is the value of \(\frac{{ta{n^2}{{60}^0} - 2si{n^2}{{45}^0}}}{{cos{{24}^0}cos{{37}^0}coses{{53}^0}cos{{60}^0}cosec{{66}^0} + si{n^2}{{60}^0}}}\)?
\(1\frac{3}{{5}}\)
SSC CHSL 2021224)If tan θ = 15, then what is the value of sec θ?
√226
SSC CHSL 2021225)Solve for x?
sin2x - 4 sin x + 3 = 0, 0 ≤ x ≤ π/2
π/2
SSC CHSL 2021226)If \(cosθ = \rm \frac{2}{3}\) , then 2 sec2θ + 2 tanθ - 6 equals:
1
SSC CHSL 2021227)If A lies between 45° and 540°, and sin A = 0.5, what is the value of A/3 in degrees?
SSC CHSL 2021228)If θ is an acute angle and sinθ = cosθ, then the value of 2 tan2θ + sin2θ - 1 is equal to:
\(\rm \frac{3}{2}\)
SSC CHSL 2021229)What number should be subtracted from
4(sin460° + cos430°) - (tan245° - cot230°) + cos245° - cosec245°+ sec260° to get 2 ?
7
SSC CHSL 2021230)Solve the following equation.
2 cos2 θ + (4 + √3)sin θ - 2(1 + √3) = 0 where θ is an acute angle.
60°
SSC CHSL 2021231)If \(\rm \frac{3 \sqrt 3 \sec θ + 4 \tan θ}{3 \tan θ + \sqrt 3 \sec θ} = 2\), 0° < θ < 90°, then the value of cos θ will be:
\(\frac{1}{2} \)
SSC CHSL 2021232)If \(tan A=\frac{{1.1}}{{6}}\) then what is the value of (4cos A - 7sin A)? Given that A is an acute angle.
\(2\frac{{41}}{{61}}\)
SSC CHSL 2021233)If \(0^0<\theta<90^0\), then \(\frac{{\left( {1 - \sin\theta } \right)\left( {\sec\theta + \tan\theta } \right)\tan\theta }}{{\left( {\tan\theta + \sec\theta + 1} \right)\left( {\cot\theta - \text{cosec} \ \theta + 1} \right)}} = ?\)
SSC CHSL 2021234)If \(\frac{4sin^2\theta + 5}{4sin^2\theta-1}\) , then the value of is:
9
SSC CHSL 2021235)If 5sin2θ = 3(1 + cosθ), 0° < θ < 90°, then the value of cosecθ + cotθ is:
\(\sqrt{\frac{7}{3}}\)
SSC CHSL 2021236)If cotθ = √2 + 1, then cosecθsecθ = ?
2√2
SSC CHSL 2021237)If \(\rm secθ = \frac{65}{63}\) and θ is an acute angle, then the value of 8(cosecθ - cotθ) is:
1
SSC CHSL 2021238)In ΔABC, if ∠B = 90°, AB = 21 cm and BC = 20 cm, then \(\rm \frac{1 \space +\space sinA \space -\space cosA}{1 \space + \space sinA \space + \space cosA} \) is equal to:
\(\frac{2}{5}\)
SSC CHSL 2021239)2 cos θ + sec θ - 2√2 = 0, where θ is an acute angle. Find the value of θ.
45°
SSC CHSL 2021240)If 2 tanx + 3 cotx = 5, then the value of 4 tan2x + 9 cot2x is:
13
SSC CHSL 2021241)What is the value of sin² 60° + tan² 45° + sec² 45° - cosec² 30° ?
\(-\frac{1}{4}\)
SSC CHSL 2021242)If 3cot2x - 7cosec2x + 7 = 0, then the value of x(0 ≤ x ≤ 90°) is:
90°
SSC CHSL 2021243)If \(8sin^2θ + 2cosθ = 5\), 0° < θ < 90°, then the value of \(tan^2θ + sec^2θ - sin^2θ\) will be:
\(\frac{305}{144}\)
SSC CHSL 2021244)If \(cosec θ = \frac{\sqrt 5}{2}\) , then what will be the value of (sec θ + tan θ - cot θ sin θ) ?
\(2 + \frac{4\sqrt 5}{5}\)
245)If in a triangle, value of length of two sides are 4cm, 5cm & angle between them is 60°, then what is the length of third side (in cm)
√21
Let third side be x cm
\(cos60^0 = {4^2 +5^2 -x^2 \over 2 \times 4\times 5} \) ; solving we get
x = √21
246)If three sides of triangle with sides AB, BC and AC are 3cm, 3cm and 4cm respectively then, what is the value of \( {sinA \over sinB}\)?
3/4
sin law \({a\over sin\space A}={b\over sin\space B}= {c\over sin\space C}\) ;
\({3 \over sinA} = {4 \over sinB}\) ; \({sinA \over sinB} = {3 \over 4} \)
247)If length of three sides of tringle AB, BC & AC are 2cm, 3cm & 4cm respectively, then the value of cos A is
11/16
applying Cosine Law, we get::
\(cos A = {4^2 +2^2 -3^2 \over 2 \times 2 \times 4} = {11 \over 16}\)
248)If tan θ = 3/4 , what is the value of Cosec θ
5/3
tan θ = P/B = 3/4 ;
By Pythagoras Theorem: –> H = 5 ;
Cosecθ = H/P = 5/3
249)If Sin θ = 4/5 , What is the value of Cos θ
3/5
Sin θ = P/H = 4/5 ;
By Pythagoras Theorem: –
\( {P^2 + B^2 = H^2}\) ;
B = 3 ;
Cos θ = B/H = 3/5;
SSC CGL 2020250)If \(12 \cos^2 \theta - \)\(2 \sin^2 \theta + \)\(3\cos \theta = 3, \) \(0^\circ < \theta < 90^\circ\), then what is the value of \(\frac{cosec \theta + \sec \theta}{\tan \theta + \cot \theta}\) ?
\(1+\sqrt3\over2\)
\(12 \cos^2 \theta - 2 \sin^2 \theta + 3\cos \theta = 3;\)
\(12 \cos^2 \theta - 2(1 - \cos^2 \theta) + 3\cos \theta = 3;\)
\(14 \cos^2 \theta + 3\cos \theta = 5; \) Put the value of \( \theta = 60^0\),
\(14 \cos^2 60^0 + 3\cos 60^0 = 5;\)
\(14 \times \frac{1}{2} + 3 \times \frac{1}{2} = 5\);
5 = 5; ⇒ L.H.S. = R.H.S. ;
\(\frac{cosec \theta + sec \theta}{tan \theta + cot \theta} = \frac{cosec 60^0 + sec 60^0}{tan 60^0 + cot 60^0}\)
\(= \frac{\frac{2}{\sqrt3} + 2}{\sqrt3 + \frac{1}{\sqrt3}} = \frac{\frac{2 + 2\sqrt3}{\sqrt3}}{\frac{3 + 1}{\sqrt3}} = \frac{1 + \sqrt3}{2}\)
SSC CGL 2020251)The value of \(sec^6\theta-tan^6\theta-3sec^2\theta \space tan^2\theta+1\over cos^4\theta-sin^4\theta+2sin^2\theta+2\) is :
\(2\over3\)
\(sec^6\theta-tan^6\theta-3sec^2\theta \space tan^2\theta+1\over cos^4\theta-sin^4\theta+2sin^2\theta+2\) = \((sec^2\theta-tan^2\theta)^3+1\over(cos^2\theta-sin^2\theta)(cos^2\theta+sin^2\theta)+2sin^2\theta+2\) \([\because sec^2\theta-tan^2\theta=1] \)
\(= {1^3+1\over cos^2\theta-sin^2\theta+2sin^2\theta+2}={2\over 1+2}={2\over3}\)
SSC CGL 2020252)If \(5sin\theta =4\), then the value of \({sec\theta+4cot\theta\over 4tan\theta-5cos\theta}\) is :
2
\(5 \sin \theta = 4\) ; ⇒ \( \sin \theta = 4/5\) ; ⇒ \(\frac{perpendicular}{hypotenuses} = \frac{4}{5}\);
By triplet 3-4-5,
Base = 3 ;
\(cos\theta = base/hypotenuses = 3/5\) ;
\(tan\theta = perpendicular/base = 4/3\) ;
\(\frac{\sec \theta + 4 \cot \theta}{4 \tan \theta - 5 \cos \theta}
= \frac{\frac{1}{\cos \theta} + \frac{4}{\tan \theta}}{4 \tan \theta - 5 \cos \theta}\)
\(={{{1\over{3\over5}} + {4\over{4\over3}}}\over{4\times {4\over3} - 5 \times {3\over5}}}\) = 2
SSC CGL 2020253)If \(11 \sin^{2} \theta - \cos^{2} \theta + 4 \sin \theta - 4 = 0, 0^\circ < \theta < 90^\circ\), then what is the value of \(\frac{\cos 2\theta + \cot 2 \theta}{\sec 2 \theta - \tan 2 \theta}\) ?
\(12+7\sqrt3\over6\)
\(11 \sin^{2} \theta - \cos^{2} \theta + 4 \sin \theta - 4 = 0\);
\(11 \sin^{2} \theta - (1 - \sin^{2} \theta) + 4 \sin \theta - 4 = 0\);
\(12 \sin^{2} \theta + 4 \sin \theta - 5 = 0\);
\(12 \sin^{2} \theta + 10 \sin \theta - 6\sin \theta- 5 = 0\);
\(2\sin \theta(6 \sin\theta + 5) -1(6 \sin\theta + 5) = 0\);
\((2\sin \theta - 1)(6 \sin\theta + 5) = 0\);
\(For 0^\circ < \theta < 90^\circ,\)
\(\sin \theta = 1/2\); \( \theta = 30^0\);
\(\frac{\cos 2\theta + \cot 2 \theta}{\sec 2 \theta - \tan 2 \theta}\)
On putting the value of \(\theta\),
\(\frac{\cos 2\times 30 + \cot 2 \times 30}{\sec 2 \times 30 - \tan 2 \times 30}\); ⇒ \(\frac{\cos 60 + \cot 60}{\sec60 - \tan 60}\); ⇒ \(\frac{\frac{1}{2} + \frac{1}{\sqrt3}}{2 - \sqrt3}\); ⇒\( \frac{2 + \sqrt3}{2\sqrt3(2 - \sqrt3)}\); ⇒ \(\frac{2 + \sqrt3}{4\sqrt3- 6}\); ⇒
\(\frac{2 + \sqrt3}{4\sqrt3- 6} \times \frac{4\sqrt3 + 6}{4\sqrt3 + 6}\); ⇒\( \frac{(2 + \sqrt3)(4\sqrt3 + 6)}{(4\sqrt3)^2- 6^2} \) ; ⇒
\(\frac{8\sqrt3 + 12 + 12 + 6\sqrt3}{12} \); ⇒ \(\frac{12 + 7\sqrt3}{6} \)
SSC CGL 2020254)What is the value of \(\frac{cosec(78^\circ + \theta) - \sec(12^\circ - \theta) - \tan(67^\circ + \theta) + \cot(23^\circ - \theta)}{\tan 13^\circ \tan37^\circ \tan45^\circ \tan53^\circ \tan77^\circ}\) ?
0
\(\frac{cosec(78^\circ + \theta) - \sec(12^\circ - \theta) - \tan(67^\circ + \theta) + \cot(23^\circ - \theta)}{\tan 13^\circ \tan37^\circ \tan45^\circ \tan53^\circ \tan77^\circ}\) = \(-tan(67^0+\theta)+tan(67^0+\theta)\over(tan13^0.cot13^0)(tan37^0.cot37^0).tan45^0\) \([\because tan(90^0-\theta)=cot\theta]\)
= \(0-0\over1\times1\times1\) = 0
SSC CGL 2020255)If \(5\space cos\theta-12\space sin\theta=0\),then what is the value of \({1+sin\space \theta +cos\space \theta\over 1-sin\space \theta+cos\space\theta}\) ?
\(3\over2\)
\(5 \cos \theta - 12 \sin \theta = 0\);
\(tan \theta = \frac{5}{12}\);
We know that \( tan \theta = \frac{perpendicular}{base}\) so,
By the triplet 5-12-13, Hypotenuse = 13 ;
\(sin \theta = \frac{5}{13}\); \(cos \theta = \frac{12}{13}\);
\(\frac{1 + \sin \theta + \cos \theta}{1 - \sin \theta + \cos \theta} = \frac{1 + \frac{5}{13} + \frac{12}{13}}{1 - \frac{5}{13} + \frac{12}{13}} = \frac{30}{20} = \frac{3}{2}\)
SSC CGL 2016256)If θ be positive acute angle and \(5cosθ + 12sinθ = 13\), then the value of cosθ is
SSC CGL 2016257)If \(tanθ + cotθ = 5\), then the value of \(tan2θ + cot2θ \) is
23
SSC CGL 2020258)The value of \(cos\space 0^0\space cos\space 30^0\space cos \space 45^0\space cos\space 60^0\space cos\space 90^0\) is:
0
\(cos\space 0^0\space cos\space 30^0\space cos \space 45^0\space cos\space 60^0\space cos\space 90^0=1\times {\sqrt3\over2}\times{1\over\sqrt2}\times{1\over2}\times0\) = 0
SSC CGL 2020259)If \(tan\space \theta-cot\space \theta = cosec\space \theta\), \(0^0<\theta<90^0\), then what is the value of \({2tan\space \theta-cos\space \theta\over \sqrt3cot\space \theta+sec\space \theta}\) ?
\(4\sqrt3-1\over6\)
Click to Watch Video SolutionConvert all the trigonometric function in sin & cos.
\(\therefore {sin\theta\over cos\theta}-{cos\theta\over sin\theta }={1\over sin\theta}\) ; ⇒ \(sin^2\theta- cos^2\theta= cos\theta\); ⇒ \(2\space cos^2\theta+cos\theta-1=0\);
Solving we get \(\theta = 60^0\);
Put \(\theta = 60^0\) in given equation we get \(4\sqrt3-1\over6\)
SSC CGL 2020260)The value of \(\sqrt {tan^2 60^0+sin90^0}-2 \space tan 45^0\) is :
0
\(\sqrt {tan^2 60^0+sin90^0}-2 \space tan 45^0=\sqrt{(\sqrt{3})^2+1}-2\times1=0\)
SSC CGL 2020261)If x cosA - y sinA = 1 and x sinA + y cosA = 4, then the value of \(17x^2+17y^2\) is:
289
Assume A = \(0^0\); \(\therefore x cos0^0-ysin0^0=1\); ⇒ x = 1; \(x sin0^0+y cos0^0=4\); y = 4; then \(17x^2+17y^2= 17+17\times(4)^2=289\)
SSC CGL 2020262)Solve the following :
\({sin40^0\over cos50^0}+{cosec50^0\over sec40^0}-4cos50^0cosec40^0\)
-2
\({sin40^0\over cos50^0}+{cosec50^0\over sec40^0}-4cos50^0cosec40^0= {sin(90^0-50^0)\over cos 50^0}+{cosec(90^0-40^0)\over sec40^0}-4cos(90^0-40^0)cosec40^0\)= \({cos 50^0\over cos50^0}+{sec40^0\over sec40^0}-4sin40^0cosec40^0= 1+1-4 = -2\) \([\because sin\theta.cos\theta=1]\)
SSC CGL 2020263)If \((2\space sin A+cosecA)=2\sqrt2\), where 0° < A < 90°, then the value of \(2(sin^4A+cos^4A)\) is:
1
\((2\space sin A+cosecA)=2\sqrt2\) ; ⇒ \((2\space sin A+{1\over sinA})=2\sqrt2\); ⇒ \(2sin^2A-2\sqrt2\space sinA+1=0\); ⇒ \(sinA={1\over\sqrt2}=sin45^0\); ⇒ \(A=45^0\); \(\therefore 2(sin^4A+cos^4A)= 2(sin^445^0+cos^445^0)=2({1\over4}+{1\over4})=2\times{1\over2}=1\)
SSC CGL 2020264)Solve the following. \(sin0^0\space sin30^0\space sin 45^0\space sin60^0\space sin90^0=?\)
0
\(sin0^0\space sin30^0\space sin 45^0\space sin60^0\space sin90^0=0\times{1\over2}\times{1\over\sqrt2}\times{\sqrt3\over2}\times1=0\)
SSC CGL 2020265)The value of \({sin30^0sin60^0\over cos60^0cos30^0}-tan45^0\) is :
0
\({{sin30^0sin60^0\over cos60^0cos30^0}-tan45^0}= {{1\over2}\times{\sqrt3\over2}\over{1\over2}\times{\sqrt3\over2}}-1 = 0\)
SSC CGL 2020266)If \((cos^2\theta-1)(1+tan^2\theta)+2tan^2\theta = 1\), \(0^0\leq\theta\leq90^0\) then \(\theta\) is :
\(45^0\)
\((cos^2\theta-1)(1+tan^2\theta)+2tan^2\theta = 1\) ; ⇒ \(-sin^2\theta.sec^2\theta+2 tan^2\theta=1\); ⇒ \({-sin^2\theta\over cos^2\theta}+2 tan^2\theta=1\); ⇒ \(-tan^2\theta+2 tan^2\theta=1\); \(tan^2\theta=1=tan^245^0\); \(\theta = 45^0\)
SSC CGL 2020267)If A lies in third quadrant, and 20 tan A = 21, then the value of \(\frac{5 \sin A - 2 \cos A}{4 \cos A - \frac{5}{7} \sin A}\) is:
1
20 tan A = 21; ⇒tan A = 21/20;
\(\frac{5 \sin A - 2 \cos A}{4 \cos A - \frac{5}{7} \sin A}
=\frac{\cos A(5 \frac{\sin A}{\cos A} - 2)}{\cos A(4 - \frac{5\sin A}{7\cos A})}
=\frac{5\tan A - 2}{4 - \frac{5}{7} \tan A}\);
On put the value of tan A,
\(= \frac{5 \times \frac{21}{20}- 2}{4 - \frac{5}{7} \times \frac{21}{20}}
= 1\)
SSC CGL 2020268)if 0 < A, B <\( 45^\circ\), \(\cos(A + B) = \frac{24}{25}\) and \(\sin(A - B) = \frac{15}{17}\), then \(\tan 2A\) is:
\(416\over87\)
tan 2A = tan((A + B) + (A - B)) =\(\frac{tan(A + B) + tan(A + B)}{1 - tan(A + B)tan(A + B)}\) ---(1);
\((\because tan(a + b) = \frac{tana + tanb}{1 - tana.tanb})
\); \(tan(A + B) = \frac{sin(A + B)}{cos(A - B)}\); ⇒
\(tan(A + B) = \frac{\sqrt{1 - cos^2(A + B)}}{cos(A + B)}\);⇒
\(tan(A + B) = \frac{\sqrt{49/25}}{(24/25)} = 7/24
\); \( tan(A - B) = \frac{sin(A - B)}{cos(A - B)}\);⇒
\(tan(A - B)= \frac{sin(A - B)}{\sqrt{1 - sin^2(A - B)}}\);⇒ \( tan(A - B) = \frac{15/17}{\sqrt{1 - (15/17)^2}}\);⇒
\(tan(A - B) = \frac{15/17}{\sqrt{64/17)^2}} = 15/8\);
From eq(1),
\(=\frac{\frac{7}{24} +\frac{15}{8}}{1 - \frac{7}{24}.\frac{15}{8}}
=\frac{416}{87}\)
SSC CGL 2020269)The value of \(4[{(1-secA)^2+(1+secA)^2\over1+sec^2A}]\) is :
8
\(4[{(1-secA)^2+(1+secA)^2\over1+sec^2A}]=4[{(1+sec^2A-2secA+1+sec^2A+2secA)\over1+sec^2A}]=4[{(2+2sec^2A)\over1+sec^2A}]= 4\times2[{(1+sec^2A)\over1+sec^2A}]= 8\)
SSC CGL 2020270)In the given figure, \(cos\theta\) is equal to:
\(5\over13\)
\(cos\theta={PR\over PQ}\); \(PR=\sqrt{PQ^2-QR^2} =\sqrt{13^2-12^2}=5\); \(\therefore cos\theta={5\over13}\)
SSC CGL 2020271)What is the value of \(sin30^0+cos30^0-tan45^0\)?
\(\sqrt3-1\over2\)
\(sin30^0+cos30^0-tan45^0={1\over2}+{\sqrt3\over2}-1 ={\sqrt3-1\over2}\)
SSC CGL 2020272)The value of \({1-2sin^2\theta.cos^2\theta\over sin^4\theta+cos^4\theta}-1\) is :
0
\({1-2sin^2\theta.cos^2\theta\over sin^4\theta+cos^4\theta}-1={1-2sin^2\theta.cos^2\theta\over (sin^2\theta+cos^2\theta)^2-2sin^2\theta.cos^2\theta}-1 = {1-2sin^2\theta.cos^2\theta\over 1-2sin^2\theta.cos^2\theta}-1=1-1 = 0\)
SSC CGL 2020273)If \(3sec^2\theta+tan\theta=7\), \(0^0<\theta<90^0\), then the value of \(cosec2\theta+cos\theta\over sin2\theta+cos\theta\) is:
\(2+\sqrt2\over4\)
\(3sec^2\theta+tan\theta=7\); ⇒ \(3(1+tan^2\theta)+tan\theta=7\); Solving we get \(\theta =45^0\);
\(\because\) Expression = \(cosec2\theta+cos\theta\over sin2\theta+cos\theta\) = \(2+\sqrt2\over4\)
SSC CGL 2020274)If \({sinA+cosA\over cosA}={17\over12}\) then the value of \(1-cosA\over sinA\) is:
\(1\over5\)
\({sinA\over cosA}+{cosA\over cosA} ={17\over12}\); ⇒ \(tanA+1 ={17\over 12}\); ⇒ tanA = \(5\over12\); Calculate \({1-cosA\over sinA }={1\over5}\)
SSC CGL 2020275)If \(5 cos^2\theta+1 =3sin^2\theta\), where, \(0^0<\theta<90^0\), then what is the value of \(tan\theta + sec\theta \over cot\theta + cosec\theta\)?
\(3+2\sqrt3\over3\)
\(5 cos^2\theta+1 =3sin^2\theta\); ⇒\(5(1-sin^2\theta)+1 =3sin^2\theta\); ⇒\(sin\theta = {\sqrt3\over2} = sin60^0\); \(\theta = 60^0\);
So \(tan\theta + sec\theta \over cot\theta + cosec\theta\) = \({tan60^0 + sec60^0 \over cot60^0 + cosec60^0 }= {3+2\sqrt3\over 3}\)
SSC CGL 2020276)Solve the following. \(({sin27^0\over cos63^0})-({cos27^0\over sin63^0})^2\)
0
\(({sin27^0\over cos63^0})-({cos27^0\over sin63^0})^2=[{sin27^0\over cos(90-27^0)}]-[{cos27^0\over sin(90-27^0)}]^2\) = \([{sin27^0\over sin27^0}]-[{cos27^0\over cos27^0}]^2 = 1 - (1)^2 =0\)
SSC CGL 2020277)If \(cot\theta+tan\theta=2sec\theta\), where \(0^0<\theta<90^0\), then the value of \(tan2\theta-sec\theta\over cot2\theta+cosec\theta\) is :
\(2\sqrt3-1\over11\)
\(cot\theta+tan\theta=2sec\theta\); \({cos\theta\over sin\theta}+{sin\theta\over cos\theta}={2\over cos\theta}\); \({cos^2\theta+sin^2\theta\over sin\theta cos\theta}={2\over\cos\theta}\); \(sin\theta={1\over2} =sin30^0\); so put \(\theta=30^0\);
\({tan 60^0-sec30^0\over cot60^0+cosec30^0}= {2\sqrt{3}-1\over11}\)
SSC CGL 2020278)The value of \((cosec30^0-tan45^0)\)\(cot60^0tan30^0\) is :
\(1\over3\)
\((cosec30^0-tan45^0)cot60^0tan30^0 = (2-1){1\over \sqrt3}\times {1\over \sqrt3}={1\over3}\)
SSC CGL 2020279)The value of \({3(1-2sin^2x)}\over{cos^2x-sin^2x}\) is :
3
\({{3(1-2sin^2x)}\over{cos^2x-sin^2x}}={ {3(1-2sin^2x)}\over{1-sin^2x-sin^2x}}\) =\({{3(1-2sin^2x)}\over(1-2sin^2x)}= 3\)
SSC CGL 2020280)The value of (\(cos10^0 \) \(cos 30^0\) \(cos 50^0\) \(cos 70^0\) \(cos 90^0\)) is :
0
\(cos 10^0 cos 30^0 cos 50 ^0 cos 70^0 cos 90^0= cos10^0. cos30^0. cos50^0. cos70^0\times0 = 0\).
SSC CGL 2020281)The value of \((cosecA+cotA+1)\)\((cosec A-cot A+1)\)\(-2 cosec A\) is:
2
(cosecA + cotA + 1)(cosecA − cotA + 1) − 2cosecA = \((cosecA + 1)^2 − (cotA)^2 − 2cosecA\); { \(a^2-b^2= (a+b)(a-b)\)}; = \(cosec^2A +1+2cosecA-cot^2A-2cosecA\)\((cosec^2A - cot^2A = 1)\) = 1 + 1 = 2
SSC CGL 2020282)If, \(5cot\theta=3\), find the value of \(6sin\theta-3cos\theta\over7sin\theta+3cos\theta\) .
\(21\over44\)
\(5cot\theta=3\); \(cot\theta={3\over5}\); (dividing numerator and denominator by \(sin\theta\)); \({6sin\theta-3cos\theta\over7sin\theta+3cos\theta}={{{6sin\theta\over sin\theta}-{3cos\theta\over sin \theta}}\over{{7sin\theta\over sin\theta}+{3cos\theta\over sin \theta}}}\)= \({6-3cot\theta\over7+3cot\theta}={6-3\times\frac{3}{5}\over7+3\times\frac{3}{5}}\) =\(21\over44\)
SSC CGL 2020283)Solve the following : \({2sin22^0\over cos68^0}-\)\({2cot75^0\over 5 tan15^0}\)\(-{8tan45^0tan20^0tan40^0tan50^0tan70^0\over5}\)
0
\({2sin22^0\over cos68^0}-{2cot75^0\over 5 tan15^0}-{8tan45^0tan20^0tan40^0tan50^0tan70^0\over5}\) =\({2sin22^0\over cos(90-22)^0}-{2cot(90-15)^0\over 5 tan15^0}-{8tan45^0tan(90-70)^0tan(90-40)^0tan50^0tan70^0\over5}\) = \({2sin22^0\over sin22^0}-{2cot75^0\over 5 cot75^0}-{8tan20^0cot20^0tan40^0cot40^0\over5}\) = \(2-{2\over5}-{8\over5}\) = 0
SSC CGL 2019284)The value of (tan29°cot61°-cosec261°)+cot254°-sec236°+(sin21°+sin23°+sin25°+.....+sin289°) is:
\(20\frac{1}{2}\)
\((tan^2 29^0- sec^229^0)+tan^236^0-sec^236^0+(sin^21^0+sin^23^0+sin^25^0+.....+cos^21^0)\)
use Identity
\((sec^2 29^0- tan^229^0) = 1\); \((sin^2 Q^0+ cos^2Q^0) = 1\)
\((Sin^245^0 = {1\over 2})\)
SSC CGL 2019285)The value of \(({sinA\over 1-cosA}+{1-cosA\over sinA})\)\(\div\)\(({cot^2A\over 1+cosecA}+1)\) is :
2
Use Hit & Trial Method, put A = 45° and match option
SSC CGL 2019286)The value of \(\sqrt{cosec\theta-cot\theta\over cosec\theta+cot\theta}\)\(\div {sin\theta\over1+cos\theta}\) is equal to :
SSC CGL 2019287)The value of \(sin(78^0+\theta)-cos(12^0-\theta)+(tan^270^0-cosec^220^0)\over sin25^0cos65^0+cos25^0sin65^0\) is :
SSC CGL 2019288)If \({sin^2\theta-3sin\theta+2\over cos^2}=1\), where \(0^0<\theta<90^0\), then what is the value of \((cos2\theta+sin3\theta\)\(+cosec2\theta)\) ?
\(9+4\sqrt3\over6\)
SSC CGL 2019289)If \(3(cot^2\theta-cos^2\theta)=cos^2\theta\), \(0^0<\theta<90^0\), then the value of \((tan^2\theta+cosec^2\theta+sin^2\theta)\) is :
SSC CGL 2019290)\((sec\theta-tan\theta)^2(1+sin\theta)^2\div sin^2\theta=?\)
\(cot^2\theta\)
SSC CGL 2019291)The value of \((tan^2\theta+cot^2\theta-sec^2\theta cosec^2\theta)\) is equal to :
-2
SSC CGL 2019292)The value of \(sec\theta(1-sin\theta)(sin\theta+cos\theta)(sec\theta+tan\theta)\over sin\theta(1+tan\theta)+cos\theta(1+cot\theta)\) is equal to :
SSC CGL 2019293)If \({1+sin\theta\over1-sin\theta}={p^2\over q^2}\), then \(sec\theta\) is equal to :
\({1\over2}({q\over p}+{p\over q})\)
SSC CGL 2019294)If \({sin\theta\over1+cos\theta}+\)\({1+cos\theta\over sin\theta}\)\(={4\over \sqrt3}\), \(0^0<\theta<90^0\), then the value of \((tan\theta+sec\theta)^{-1}\) is :
\(2-\sqrt3\)
\({{sin^2θ+1+cos^2θ+2cosθ} \over sinθ(1+cosθ)}={4 \over √ 3}\)
⇒ \({sinθ = }{√ 3 \over 2}\), θ = 60°
SSC CGL 2019295)The value of \({sin\theta+cos\theta-1\over sin\theta-cos\theta+1}\times\)\({ tan^2\theta(cosec^2\theta-1)\over sec\theta-tan\theta}\) is :
SSC CGL 2019296)The value of \({(sin\theta-cos\theta)(1+tan\theta+cot\theta)\over1+sin\theta cos\theta} = ?\)
\(sec\theta-cosec\theta\)
\({(sin\theta-cos\theta)(1+{sin\theta \over cos\theta}+ {cos\theta \over sin\theta})\over1+sin\theta cos\theta}\)
\((sinθ-cosθ)(sinθcosθ + sin^2θ +cos^2θ) \over (1+sinθcosθ)(sinθcosθ)\)
\(sec\theta-cosec\theta\)
SSC CGL 2019297)The value of \(sin^264^0+\)\(cos64^0sin26^0+\)\(2cos43^0cosec47^0\) is :
3
SSC CGL 2019298)The value of \({2(sin^6\theta+cos^6\theta)-3(sin^4\theta+cos^4\theta)}\over{cos^4\theta-sin^4\theta-2cos^2\theta}\) is :
1
SSC CGL 2019299)The value of \({sec^2\theta\over cosec^2\theta}+\)\( {cosec^2\theta\over sec^2\theta}-\)\((sec^2\theta+cosec^2\theta)\) is :
-2
SSC CGL 2019300)The value of \((1+cot\theta-cosec\theta) (1+cos\theta+sin\theta)\)\(sec\theta = ?\)
2
SSC CGL 2019301)If \(2cos^2\theta+3sin\theta=3\), where \(O^0<\theta<90^0\), then what is the value of \(sin^22\theta+cos^2\theta +\)\(tan^22\theta +cosec^22\theta\) ?
\(35\over6\)
SSC CGL 2019302)The value of \(cosec(67^0 +\theta)-\)\(sec(23^0- \theta) +\)\(cos15^0cos35^0cosec55^0 \)\(cos60^0cosec75^0\) is :
\(1\over2\)
\((67^0 +\theta)-sec(23^0- \theta) \)\(+cos15^0cos35^0cosec55^0cos60^0cosec75^0\)= \((67^0 +\theta)-cosec(67^0+\theta) +\)\(cos15^0cos35^0sec35^0cos60^0sec15^0 \)\(= cos60^0={1\over2}\).
SSC CGL 2019303)If \(sec\theta + tan\theta =p\), (p >1) then \({cosec\theta +1\over cosec\theta-1 } = ?\)
\(p^2\)
\(sec\theta+tan\theta=p\)________(1) \(sec\theta-tan\theta={1\over p}\)_______(2); from eq (1) and (2),\(sec\theta={p^2+1\over2p}\); \(cos\theta= {2p\over p^2+1}\); \(sin\theta={p^2-1\over p^2+1}\); Now calculate \(cosec\theta+1\over cosec\theta-1\) it will come out to be \(p^2\).
SSC CGL 2019304)If 5 \({sin\theta}\) - 4 \(cos\theta\) = 0, 0º < \(\theta\) < 90º , then the value of \({5 sin\theta- 2cos\theta \over 5sin\theta + 3cos\theta}\) is :
\({2 \over 7}\)
\(5\sin\theta-4\cos\theta=0,0^{\circ}<\theta<90^{\circ}\)
\(= 5\sin\theta = 4\cos\theta\)
\(= \tan\theta = \frac{4}{5};\)
Now,
\(\frac{5\sin\theta-2\cos\theta}{5\sin\theta+3\cos\theta}
= \frac{\cos\theta(5\tan\theta-2)}{\cos\theta(5\tan\theta+3)}
= \frac{5\tan\theta-2}{5\tan\theta+3}
= \frac{5\times \frac{4}{5} -2}{5\times \frac{4}{5} +3}
= \frac{4 -2}{4 +3}
= \frac{2}{7}\)
SSC CGL 2019305)\(\sqrt{cot\theta+cos\theta\over cot\theta-cos\theta}\) is equal to :
\(sec\theta+tan\theta\)
\(\sqrt{\frac{\cot\theta+\cos\theta}{\cot\theta-\cos\theta}}\)
\(= \sqrt{\frac{cos\theta(\frac{1}{sin\theta}+1)}{cos\theta(\frac{1}{sin\theta} - 1)}}
= \sqrt{\frac{1 + sin\theta}{1 - sin\theta} \times \frac{1 + sin\theta}{1 + sin\theta}}\)
\(= \sqrt{\frac{(1 + sin\theta)^2}{1 - sin^2\theta}}
= \frac{1 + sin\theta}{cos\theta}
= sec\theta + tan\theta\)
SSC CGL 2019306)\(({1-tan\theta\over 1- cot\theta})^2+1=?\)
\(sec^2\theta\)
\(\left(\frac{1-\tan\theta}{1-\cot\theta}\right)^2+1;\)
\((\frac{1-\tan\theta}{1-\frac{1}{tan\theta}})^2+1;\)
\((\frac{1-\tan\theta}{\frac{tan\theta - 1}{tan\theta}})^2+1;\)
\((-tan\theta)^2 + 1;\)
\(tan^2\theta + 1 = sec^2\theta\)
SSC CGL 2019307)\({(1+cos\theta)^2+sin^2\theta\over (cosec^2\theta-1)sin^2\theta}=?\)
\(2sec\theta(1+sec\theta)\)
\(\frac{\left(1+\cos\theta\right)^2+\sin^2\theta}{\left(\text{coec}^2\theta-1\right)\sin^2\theta}\)
\(=\frac{1 + \cos^2\theta + 2\cos\theta +\sin^2\theta}{1 -\sin^2\theta}
=\frac{2 + 2\cos\theta }{\cos^2\theta}
=2(\sec^2\theta+ \sec\theta)
=2\sec\theta(1 + \sec\theta)\)
SSC CGL 2019308)What is the value of \(cosec(65^0+\theta)-\)\(sec(25^0-\theta)+\)\(tan^220^0-\)\(cosec^270^0\) ?
-1
\(\ cosec(65^\circ + \theta) - \sec(25^\circ - \theta) + \tan^2 20^\circ - \ cosec^2 70^\circ =\ cosec(65^\circ + \theta) - \sec(90 - (65^\circ + \theta)) + \tan^2 20^\circ - \ cosec^2 (90 - 20)\)
\(=\ cosec(65^\circ + \theta) - \ cosec(65^\circ + \theta) + \tan^2 20^\circ - \sec^2 20^\circ = \tan^2 20^\circ - \sec^2 20^\circ = -(\sec^2 20^\circ -\tan^2 20^\circ ) = -1\)
SSC CGL 2019309)If \(\theta\) lies in the first quadrant and \({(cos^2\theta-sin^2\theta)}={1\over 2}\), then the value of \({tan^22\theta+sin^23\theta}\) is :
4
\(\cos^2 \theta - \sin^2 \theta = \frac{1}{2}\);
\(\cos^2 \theta - (1 - \cos^2 \theta) = \frac{1}{2};\)
\(2\cos^2 \theta - 1 = \frac{1}{2}; \) \( 2\cos^2 \theta = \frac{3}{2};\)
\(\cos^2 \theta = \frac{3}{4};\)
\(\cos \theta = \frac{\sqrt{3}}{2};\)
\(\theta = 30^0;\)
Now,
\(\tan^2 2\theta + \sin^2 3\theta; = \tan^2 2\times30^0 + \sin^2 3\times30^0 = \tan^2 60^0+ \sin^2 90^0 = 3 + 1 = 4\)
SSC CGL 2019310)The value of \({(cos9^0 + sin81^0)(sec9^0 + cosec81^0)} \over {sin56^0sec34^0 + cos25^0cosec65^0}\) is :
2
\(\frac{(\cos 9^\circ + \sin 81^\circ)(\sec 9^\circ + \ cosec 81^\circ)}{\sin 56^\circ sec 34^\circ + \cos 25^\circ \ cosec 65^\circ} = \frac{(\cos 9^\circ + \sin(90 - 9))(\sec 9^\circ + \ cosec(90 - 9))}{\sin 56^\circ sec(90 - 56) + \cos 25^\circ \ cosec(90 - 25)} = \frac{(\cos 9^\circ + \cos 9^\circ)(\sec 9^\circ + \sec 9^\circ)}{\sin 56^\circ cosec 56^\circ + \cos 25^\circ \sec 25^\circ} = \frac{(2\cos 9^\circ)(2 \sec 9^\circ)}{1 + 1} = \frac{4}{2} = 2\)
SSC CGL 2019311)\({(2sinA)(1+sinA)}\over{1+sinA+cosA}\) is equal to :
1 + sin A - cos A
\(\frac{(2 \sin A)(1 + \sin A)}{1 + \sin A + \cos A}; = \frac{(2 \sin A + 2\sin^2 A)}{1 + \sin A + \cos A} = \frac{(2 \sin A + 2 - 2\cos^2 A)}{1 + \sin A + \cos A}\)
\((\because \sin^2 A + \cos^2 A = 1);\)
=\( \frac{(2 \sin A + 1 + \sin^2 A + \cos^2 A - 2\cos^2 A)}{1 + \sin A + \cos A} = \frac{((\sin A + 1)^2 - \cos^2 A)}{1 + \sin A + \cos A}\)
\((\because (a)^2 - (b)^2 = (a + b)(a - b));\)
=\( \frac{(1 + \sin A + \cos A)(1 + \sin A - \cos A)}{1 + \sin A + \cos A} = (1 + \sin A - \cos A)\)
SSC CGL 2019312)The value of the expression (\({cos^6\theta+sin^6\theta-1}\)) (\({tan^2\theta+cot^2\theta+2}\)) is :
-3
\((\cos^6 \theta + \sin^6 \theta - 1)(\tan^2 \theta + \cot^2 \theta + 2);\)
Assume any value of \theta,
Let the \( \theta\) be \(45^0\)
= \((\cos^6 45^0 + \sin^6 45^0- 1)(\tan^2 45^0 + \cot^2 45^0 + 2) = ((\frac{1}{\sqrt{2}})^6 + (\frac{1}{\sqrt{2}})^6 - 1)(1 + 1 + 2) = ((\frac{1}{8}) + (\frac{1}{8}) - 1)(4) = \frac{3}{4} \times 4 = -3\)
SSC CGL 2019313)If sin\({\theta}\) = \( { \sqrt3}\) cos\({\theta}\), 0°< \({\theta}\) < 90º , then the value of (\(2sin^2{\theta}+sec^2{\theta}\)\(+sin{\theta}sec{\theta}+cosec{\theta}\)) is:
\( {33+10\sqrt3 \over 6}\)
\(\sin \theta = \sqrt{3} \cos \theta, 0^\circ < \theta < 90^\circ \Rightarrow \frac{\sin \theta}{\cos \theta} = \sqrt{3} \Rightarrow \tan \theta = \sqrt{3} \Rightarrow \theta = 60 \),
put 60 degree in the question and the answer will come out to be A
SSC CGL 2020314)In a right triangle shown in the figure, what is the value of \(cosec\theta\) ?
13/5
\(cosec\theta = {H /P}\)
SSC CGL 2020315)If \(6tan\theta-5\sqrt3sec\theta+12cot\theta=0\) where, \(0 <\theta< 90\), then value of \(cosec\theta +sec\theta\) is:
\({2(3+\sqrt3) \over 3}\)
since, \(0 <\theta< 90\), it is always advisable in competitive exams to use \(\theta = 30, 45 and 60\) In the instant question \(\theta = 60 \) will satisfy the condition \(6tan\theta-5\sqrt3sec\theta+12cot\theta=0\) , therefore put \(\theta = 60 \) in \(cosec\theta +sec\theta\) and calulate answer.