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UPSC NDA II 2025 Mathematics Paper

Question Numbers: 25-26

For the following two (02) items:

Let sin A + sin

B = p and cos A + cos B

= q

Question : 25 of 120

Marks: +1, -0

What is p/q equal to?

Solution:

Concept:

Use sum-to-product identities to transform the sum of sines and cosines into products, then simplify the ratio.

Explanation:

Given

p = \sin A + \sin B

and

q = \cos A + \cos B

Apply the identities:

\sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)

\cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)

\frac{p}{q} = \frac{2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)}{2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)}

Cancel the common factor

2 \cos\left(\frac{A-B}{2}\right)

This gives

\frac{p}{q} = \frac{\sin\left(\frac{A+B}{2}\right)}{\cos\left(\frac{A+B}{2}\right)} = \tan\left(\frac{A+B}{2}\right)

Answer:

Option C:

\tan\left(\frac{A+B}{2}\right)

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