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UPSC NDA 2 2019 Mathematics Solved Paper

Question : 68 of 120

Marks: +1, -0

What is

cot(\frac{A}{2})-tan(\frac{A}{2})

equal to?

Solution:

Concept:

Use double-angle identities for cosine and sine to simplify the expression.

Explanation:

Start with

\cot(\frac{A}{2}) - \tan(\frac{A}{2})

Rewrite as

\dfrac{\cos(\frac{A}{2})}{\sin(\frac{A}{2})} - \dfrac{\sin(\frac{A}{2})}{\cos(\frac{A}{2})}

Combine into a single fraction:

\dfrac{\cos^2(\frac{A}{2}) - \sin^2(\frac{A}{2})}{\sin(\frac{A}{2})\cos(\frac{A}{2})}

The numerator is

\cos A

because

\cos 2\theta = \cos^2\theta - \sin^2\theta

Here

\theta = \frac{A}{2}

, so

\cos A = \cos^2(\frac{A}{2}) - \sin^2(\frac{A}{2})

The denominator is

\sin(\frac{A}{2})\cos(\frac{A}{2}) = \frac{1}{2} \sin A

using

\sin 2\theta = 2\sin\theta\cos\theta

Thus the expression becomes

\dfrac{\cos A}{\frac{1}{2}\sin A} = 2\dfrac{\cos A}{\sin A} = 2\cot A

Answer:

2\cot A

, which corresponds to option D.

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