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UPSC CDS 1 2021 Math Paper

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Question : 82 of 100

Marks: +1, -0

If

p = \sec \theta - \tan \theta

q = \csc \theta + \cot \theta

p+q(p-1)

equal to?

-1
0
1
2

Solution: 👈: Video Solution

p = \sec \theta - \tan \theta = \frac{1-\sin \theta}{\cos \theta}

and

q = \csc \theta + \cot \theta = \frac{1+\cos \theta}{\sin \theta}

p+q(p-1)

= (p-q) + pq

= \left(\frac{1-\sin \theta}{\cos \theta} - \frac{1+\cos \theta}{\sin \theta}\right)

+ \left(\frac{1-\sin \theta}{\cos \theta} \times \frac{1+\cos \theta}{\sin \theta}\right)

= \frac{\sin \theta - \sin^2 \theta - \cos \theta - \cos^2 \theta}{\sin \theta \cos \theta}

+ \frac{1 - \sin \theta + \cos \theta - \sin \theta \cos \theta}{\sin \theta \cos \theta}

= \frac{\sin \theta - \cos \theta - 1}{\sin \theta \cos \theta}

+ \frac{1 - \sin \theta + \cos \theta - \sin \theta \cos \theta}{\sin \theta \cos \theta}

= \frac{-\sin \theta \cos \theta}{\sin \theta \cos \theta}

= -1

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