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UPSC CDS II 2025 Mathematics Paper

Question : 34 of 100

Marks: +1, -0

What is

(sec\theta -tan\theta )-\sqrt{\frac{1-sin\theta }{1+sin\theta }}

equal to?

Solution:

Concept:

The expression simplifies using the identity

\sqrt{\frac{1-\sin\theta}{1+\sin\theta}} = \sec\theta - \tan\theta

, which follows from multiplying numerator and denominator by

1-\sin\theta

and using

\sec^2\theta - \tan^2\theta = 1

Explanation:

Start with the given expression:

(\sec\theta - \tan\theta) - \sqrt{\frac{1-\sin\theta}{1+\sin\theta}}

We know

\sqrt{\frac{1-\sin\theta}{1+\sin\theta}} = \sec\theta - \tan\theta

for appropriate

\theta

Substitute this into the expression:

(\sec\theta - \tan\theta) - (\sec\theta - \tan\theta)

The two identical terms cancel each other, leaving

0

Thus the value is independent of

\theta

and equals zero.

Answer:

0 (option A)

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