First slide

Compound angles

Question

If $\sin A + \sin B = a$ and $\cos A + \cos B = b,$ then $\cos (A + B)$

Moderate

A

$\frac{a^{2} + b^{2}}{b^{2} - a^{2}}$
B

$\frac{2 a b}{a^{2} + b^{2}}$
C

$\frac{b^{2} - a^{2}}{a^{2} + b^{2}}$
D

$\frac{a^{2} - b^{2}}{a^{2} + b^{2}}$

Solution

We have,
$a = \sin A + \sin B, b = \cos A + \cos B$
$\Rightarrow a^{2} + b^{2} = 2 + 2 \cos (A - B)$ …(i)
and,
$\begin{array}{l} b^{2} - a^{2} = \cos 2 A + \cos 2 B + 2 \cos (A + B) \\ \Rightarrow b^{2} - a^{2} = 2 \cos (A + B) {\cos (A - B) + 1} \end{array}$
$\Rightarrow b^{2} - a^{2} = 2 \cos (A + B) (\frac{a^{2} + b^{2}}{2})$ [Using (i)]
$\Rightarrow \cos (A + B) = \frac{b^{2} - a^{2}}{a^{2} + b^{2}}$

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