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  4. If sin x+ sin y=a and cos x+ cos y=b then tan ((x+y/2))=

Q. If $\sin x+\sin y=a$ and $\cos x+\cos y=b$ then $\tan \left(\frac{x+y}{2}\right)=$

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Solution:

$\sin x +\sin y = a$
$ \Rightarrow 2 \sin \frac{x+y}{2} \cos \frac{x-y}{2}= a ; \cos x +\cos y= b $
$\Rightarrow 2 \cos \frac{x+y}{2} \cos \frac{x-y}{2}= b$
$\frac{a}{b}=\tan \frac{x+y}{2}$

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