# Question #00111

Oct 12, 2016

We have sine and cosine rule as follows

$\sin \left(A + B\right) = \sin A \cos B + \cos A \sin B$

and

$\cos \left(A + B\right) = \cos A \cos B - \sin A \sin B$

To prove sine rule using cosine rule.

Cosine rule being an identity it is valid for all real values of A and B.So we can put ($\frac{\pi}{2} + A$) in place of A in the 2nd identity of cosine and get

$\cos \left(\frac{\pi}{2} + A + B\right) = \cos \left(\frac{\pi}{2} + A\right) \cos B - \sin \left(\frac{\pi}{2} + A\right) \sin B$

$\implies - \sin \left(A + B\right) = - \sin A \cos B - \cos A \sin B$

$\implies \sin \left(A + B\right) = \sin A \cos B + \cos A \sin B$

This is the sine rule.

Similarly we can also prove it for rule of $\sin \left(A - B\right) \mathmr{and} \cos \left(A - B\right)$