# Simplify cos x - cos 2x + sin 2x - sin 3x = ?

Feb 18, 2017

Use these trig identities:
$\cos a - \cos b = - 2 \sin \left(\frac{a + b}{2}\right) \sin \left(\frac{a - b}{2}\right)$
$\sin a - \sin b = 2 \cos \left(\frac{a + b}{2}\right) \sin \left(\frac{a - b}{2}\right)$
In this case:
$\cos x - \cos 2 x = 2 \sin \left(\frac{3 x}{2}\right) \sin \left(\frac{x}{2}\right)$ (1)
$\sin 2 x - \sin 3 x = - 2 \cos \left(\frac{5 x}{2}\right) \sin \left(\frac{x}{2}\right)$ (2)
Put $\sin \left(\frac{x}{2}\right)$ into common factor, we get:
$\cos x - \cos 2 x + \sin 2 x - \sin 3 x =$
$= 2 \sin \left(\frac{x}{2}\right) \left[\sin \left(\frac{3 x}{2}\right) - \cos \left(\frac{5 x}{2}\right)\right]$