What is sinθ /1-cotθ +cosθ /1-tanθ (θ ≠ π/4) equal to?

Correct Answer is : sinθ+cosθsin θ + cos θsinθ+cosθ

Correct Answer is : sinθ+cosθsin θ + cos θsinθ+cosθ

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UPSC CDS I 2025 Mathematics Paper

Question : 26 of 100

Marks: +1, -0

What is

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

≠

π/4) equal to?

Solution:

Concept:

Rewrite cot and tan in terms of sin and cos, then simplify each fraction separately and add using a common denominator.

Explanation:

Start with

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

Replace

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

and

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

First term:

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

Since

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

, the second term becomes

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

Add the 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}

Factor numerator:

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

Cancel

\sin\theta-\cos\theta

(non-zero because

\theta\neq\pi/4

). This gives

\sin\theta+\cos\theta

Answer:

\sin\theta+\cos\theta

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