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UPSC CDS I 2025 Mathematics Paper

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Question : 26 of 100

Marks: +1, -0

What is

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

\neq

π/4) equal to?

sin θ + cos θ
sin θ − cos θ
cos θ − sin θ
− (sin θ + cos θ)

Solution:

First term:

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

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

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

Second term:

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

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

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

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

Now, add both terms:

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

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

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

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

= -(-1)(\cos\theta+\sin\theta)

= \cos\theta+\sin\theta

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