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UPSC NDA 2 2019 Mathematics Solved Paper

Question : 69 of 120

Marks: +1, -0

What is cot A + cosec A equal to?

Solution:

Concept:

Use trigonometric identities to simplify the sum of cotangent and cosecant.

Explanation:

Write in terms of sine and cosine:

\cot A + \csc A = \frac{\cos A}{\sin A} + \frac{1}{\sin A} = \frac{1+\cos A}{\sin A}

Apply half-angle identities:

1+\cos A = 2\cos^{2}\frac{A}{2}

and

\sin A = 2\sin\frac{A}{2}\cos\frac{A}{2}

Substitute:

\frac{1+\cos A}{\sin A} = \frac{2\cos^{2}\frac{A}{2}}{2\sin\frac{A}{2}\cos\frac{A}{2}} = \frac{\cos\frac{A}{2}}{\sin\frac{A}{2}} = \cot\frac{A}{2}

Thus,

\cot A + \csc A

simplifies to

\cot\frac{A}{2}

Answer:

Option B:

\cot\frac{A}{2}

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