# How do you prove  sin(a+b) + sin(a+b) = 2sinacosb ?

Mar 27, 2016

You cannot prove it. It is not always true. But it is true that $\sin \left(a + b\right) + \sin \left(a - b\right) = 2 \sin a \cos b$

#### Explanation:

$\sin \left(a + b\right) = \sin a \cos b + \cos a \sin b$.

So,

$\sin \left(a + b\right) + \sin \left(a + b\right) = 2 \sin \left(a + b\right)$

$= 2 \left(\sin a \cos b + \cos a \sin b\right)$

$= 2 \sin a \cos b + 2 \cos a \sin b$.

The last is equal to $2 \sin a \cos b$ only if $\cos a = 0$ or $\sin b = 0$

But

$\sin \left(a - b\right) = \sin a \cos b - \cos a \sin b$.

So,

$\sin \left(a + b\right) + \sin \left(a - b\right) = \left(\sin a \cos b + \cos a \sin b\right) + \left(\sin a \cos b - \cos a \sin b\right)$

$= 2 \sin a \cos b$