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The value of $\sin A + \sin B = x$ and $\cos A + \cos B = y$ then $\cos (A - B) =$

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$\frac{x^{2} + y^{2} + 2}{2}$

$\frac{x^{2} - y^{2} + 2}{2}$

$\frac{x^{2} + y^{2} - 2}{2}$

$\frac{y^{2} - x^{2} + 2}{2}$

answer is C.

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Detailed Solution

$\sin A + \sin B = x$

Squaring on both sides

$\cos A + \cos B = y$

Squaring on both sides

Adding two functions

$2 \cos (A - B) = x^{2} + y^{2} - 2$

$\cos (A - B) = \frac{x^{2} + y^{2} - 2}{2}$

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