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SSC CHSL 2023 PYQ — Question 59

SSC CHSL - 07 Aug 2023 - Shift 4 • Quantitative Aptitude • Q59

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The value of 1sinθ−cot2θ1+cosecθ\frac{1}{sin\theta}-\frac{cot^2\theta}{1+cosec\theta} is:

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Correct answer: B

Explanation

The correct answer is option 2 i.e. 1.

1sinθ−cot2θ1+cosecθ\frac{1}{sin\theta}-\frac{cot^2\theta}{1+cosec\theta}

⇒ 1sinθ\frac{1}{sin\theta} - cos2θsin2θ1+1sinθ\frac{{cos^2\theta}\over {sin^2\theta}}{1+{1\over sin\theta}}

⇒ 1sinθ\frac{1}{sin\theta} - cos2θsin2θsinθ+1sinθ\frac{{cos^2\theta}\over {sin^2\theta}}{{{sin\theta + 1} \over sin\theta}}

⇒​​​​​​​ 1sinθ\frac{1}{sin\theta} - cos2θ×sinθsin2θ(sinθ+1)\frac{{cos^2\theta \times sin\theta}}{{sin^2\theta({sin\theta + 1} )}}

⇒​​​​​​​ 1sinθ\frac{1}{sin\theta} - (1−sin2θ)×sinθsin2θ(sinθ+1)\frac{{(1 - sin^2\theta) \times sin\theta}}{{sin^2\theta({sin\theta + 1} )}}

⇒​​​​​​​ 1sinθ\frac{1}{sin\theta} - (1−sinθ)(1+sinθ)sinθ(sinθ+1)\frac{{(1 - sin\theta)(1+sin\theta) }}{{sin\theta({sin\theta + 1} )}}

⇒​​​​​​​ 1sinθ\frac{1}{sin\theta} - (1−sinθ)sinθ\frac{{(1 - sin\theta) }}{{sin\theta}}

⇒​​​​​​​ 1−1+sinθsinθ\frac{1-1+sin\theta}{sin\theta}

⇒​​​​​​​ sinθsinθ\frac{sin\theta}{sin\theta} = 1