# Using the double angle of half angle formula, how do you simplify cos^2 5theta- sin^2 5theta?

Apr 25, 2015

There is another simple way to simplify this.
${\cos}^{2} 5 x - {\sin}^{2} 5 x = \left(\cos 5 x - \sin 5 x\right) \left(\cos 5 x + \sin 5 x\right)$
Use the identities:
$\cos a - \sin a = - \left(\sqrt{2}\right) \cdot \left(\sin \left(a - \frac{\Pi}{4}\right)\right)$

$\cos a + \sin a = \left(\sqrt{2}\right) \cdot \left(\sin \left(a + \frac{\Pi}{4}\right)\right)$

So this becomes:
$- 2 \cdot \sin \left(5 x - \frac{\Pi}{4}\right) \cdot \sin \left(5 x + \frac{\Pi}{4}\right)$.

Since $\sin a \cdot \sin b = \frac{1}{2} \left(\cos \left(a - b\right) - \cos \left(a + b\right)\right)$, this equation can be rephrased as (removing the parentheses inside the cosine):

$- \left(\cos \left(5 x - \frac{\Pi}{4} - 5 x - \frac{\Pi}{4}\right) - \cos \left(5 x - \frac{\Pi}{4} + 5 x + \frac{\Pi}{4}\right)\right)$

This simplifies to:

$- \left(\cos \left(- \frac{\pi}{2}\right) - \cos \left(10 x\right)\right)$

The cosine of $- \frac{\pi}{2}$ is 0, so this becomes:

$- \left(- \cos \left(10 x\right)\right)$

$\cos \left(10 x\right)$

Unless my math is wrong, this is the simplified answer.