# How do you prove 1-cos5thetacos3theta-sin5thetasin3theta=2sin^2theta?

Sep 20, 2016

We use the identity $\cos 2 \theta = 1 - 2 {\sin}^{2} \theta$ and $\cos \left(A - B\right) = \cos A \cos B + \sin A \sin B$

#### Explanation:

We know that $\cos 2 \theta = 1 - 2 {\sin}^{2} \theta$

But as $2 \theta = 5 \theta - 3 \theta$, we have

$\cos \left(5 \theta - 3 \theta\right) = 1 - 2 {\sin}^{2} \theta$

or $\cos 5 \theta \cos 3 \theta + \sin 5 \theta \sin 3 \theta = 1 - 2 {\sin}^{2} \theta$

or $1 - \cos 5 \theta \cos 3 \theta - \sin 5 \theta \sin 3 \theta = 2 {\sin}^{2} \theta$