First slide

Compound angles

Question

If $\frac{x}{y} = \frac{\cos A}{\cos B}$ then $\frac{x \tan A + y \tan B}{x + y} =$

Moderate

A

$\frac{\sin A + \cos B}{\cos A + \sin B}$
B

$\frac{\sin A + \sin B}{\cos A \cos B}$
C

$\tan \frac{A + B}{2}$
D

$\cot \frac{A - B}{2}$

Solution

$\frac{x}{\cos A} = \frac{y}{\cos B}$ so $\frac{x \tan A + y \tan B}{x + y}$
$\begin{array}{l} = \frac{\cos A \tan A + \cos B \tan B}{\cos A + \cos B} \\ = \frac{\sin A + \sin B}{\cos A + \cos B} = \frac{2 \sin \frac{A + B}{2} \cos \frac{A - B}{2}}{2 \cos \frac{A + B}{2} \cos \frac{A - B}{2}} \\ = \tan \frac{A + B}{2} \end{array}$

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