SSC CGL 201911)\(\sqrt{cot\theta+cos\theta\over cot\theta-cos\theta}\) 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 201912)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 201913)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 201914)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 201915)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 201916)The value of \((1+cot\theta-cosec\theta) (1+cos\theta+sin\theta)\)\(sec\theta = ?\)
2
SSC CGL 201917)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 201918)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 201919)The value of \(sin^264^0+\)\(cos64^0sin26^0+\)\(2cos43^0cosec47^0\) is :
3
SSC CGL 201920)The value of \({(sin\theta-cos\theta)(1+tan\theta+cot\theta)\over1+sin\theta cos\theta} = ?\)
\(sec\theta-cosec\theta\)
\({(sin\theta-cos\theta)(1+{sin\theta \over cos\theta}+ {cos\theta \over sin\theta})\over1+sin\theta cos\theta}\)
\((sinθ-cosθ)(sinθcosθ + sin^2θ +cos^2θ) \over (1+sinθcosθ)(sinθcosθ)\)
\(sec\theta-cosec\theta\)