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SSC CGL 2022 PYQ — Question 35

SSC CGL Tier II - Quant - 3 Feb 2022 • Quantitative Aptitude • Q35

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The expression (1−sinθ+cosθ)2(1−cosθ)sec3θcosec2θ(secθ−tanθ)(tanθ+cotθ),00<θ<900\frac{(1-sin \theta+cos\theta)^2(1-cos \theta)sec^3 \theta cosec^2 \theta}{(sec \theta - tan \theta)(tan \theta + cot \theta)}, 0^0<\theta<90^0 is equal to:

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Correct answer: D

Explanation

The correct answer is option 4 i.e.  2 tan θ\theta

Shortcut = Put θ\theta = 45°.

⇒ (1−sinθ+cosθ)2(1−cosθ)sec3θcosec2θ(secθ−tanθ)(tanθ+cotθ),00<θ<900\frac{(1-sin \theta+cos\theta)^2(1-cos \theta)sec^3 \theta cosec^2 \theta}{(sec \theta - tan \theta)(tan \theta + cot \theta)}, 0^0<\theta<90^0

⇒ (1−12+12)2(1−12)2322(2−1)(1+1)\frac{(1- \frac{1}{\sqrt2}+\frac{1}{\sqrt2})^2(1-\frac{1}{\sqrt2})\sqrt2^3 \sqrt2^2 }{(\sqrt2 - 1)(1 + 1)}

⇒ 1×2−12×422−1×2\frac{1 \times \frac{\sqrt2 - 1}{\sqrt2} \times 4\sqrt2}{\sqrt2 - 1 \times 2} = 2.

Now put θ\theta = 45° in options

option 1 =  sin θ\theta = 1/2\sqrt2

option 2 =  2 cos θ\theta = 2\sqrt2

option 3 =  cot θ\theta = 2\sqrt2

option 4 =  2 tan  θ\theta = 2 ( matches with our answer).