The expression sinα + sinβ /cosα + cosβ is equal to

Correct Answer is : tan(α+β2)tan(α + β /2)tan(2α+β)

Correct Answer is : tan(α+β2)tan(α + β /2)tan(2α+β)

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UPSC NDA I 2017 Mathematics Solved Paper

Question : 47 of 120

Marks: +1, -0

The expression

\frac{sin\alpha + sin\beta }{cos\alpha + cos\beta }

is equal to

Solution:

Concept:

Use sum-to-product trigonometric identities to simplify the ratio.

Explanation:

Apply the identities:

\sin\alpha + \sin\beta = 2 \sin\left(\frac{\alpha+\beta}{2}\right) \cos\left(\frac{\alpha-\beta}{2}\right)

\cos\alpha + \cos\beta = 2 \cos\left(\frac{\alpha+\beta}{2}\right) \cos\left(\frac{\alpha-\beta}{2}\right)

Substitute into the given expression:

\frac{\sin\alpha + \sin\beta}{\cos\alpha + \cos\beta} = \frac{2 \sin\left(\frac{\alpha+\beta}{2}\right) \cos\left(\frac{\alpha-\beta}{2}\right)}{2 \cos\left(\frac{\alpha+\beta}{2}\right) \cos\left(\frac{\alpha-\beta}{2}\right)}

Cancel the common factor

2 \cos\left(\frac{\alpha-\beta}{2}\right)

= \frac{\sin\left(\frac{\alpha+\beta}{2}\right)}{\cos\left(\frac{\alpha+\beta}{2}\right)} = \tan\left(\frac{\alpha+\beta}{2}\right)

Answer:

\tan\left(\frac{\alpha+\beta}{2}\right)

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