If $\cos α + \cos β = m$ and $\sin α + \sin β = n,$ then $\sin (α + β) =$

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$\frac{m^{2} - n^{2}}{m^{2} + n^{2}}$

$\frac{n^{2} - m^{2}}{m^{2} + n^{2}}$

$\frac{2 m n}{m^{2} - n^{2}}$

$\frac{2 m n}{m^{2} + n^{2}}$

answer is C.

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Detailed Solution

$m = \cos α + \cos β = 2 \cos (\frac{α + β}{2}) \cos (\frac{α - β}{2})$

$n = \sin α + \sin β = 2 \sin (\frac{α + β}{2}) \cos (\frac{α - β}{2})$

$∴ \frac{n}{m} = \tan (\frac{α + β}{2})$
Now,

$\sin (α + β) = \frac{2 \tan (\frac{α + β}{2})}{1 + \tan^{2} (\frac{α + β}{2})} = \frac{\frac{2 n}{m}}{1 + (\frac{n^{2}}{m^{2}})} = \frac{2 m n}{m^{2} + n^{2}}$

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