# How do you use product to sum formulas to write the product cos5thetacos3theta as a sum or difference?

Apr 25, 2017

$\cos 5 \theta \cos 3 \theta = \frac{1}{2} \cos 8 \theta + \frac{1}{2} \cos 2 \theta$

#### Explanation:

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

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

Adding two, we get $\cos \left(A + B\right) + \cos \left(A - B\right) = 2 \cos A \cos B$

or $\cos A \cos B = \frac{1}{2} \left[\cos \left(A + B\right) + \cos \left(A - B\right)\right]$

Hence, $\cos 5 \theta \cos 3 \theta$

= $\frac{1}{2} \left[\cos \left(5 \theta + 3 \theta\right) + \cos \left(5 \theta - 3 \theta\right)\right]$

= $\frac{1}{2} \cos 8 \theta + \frac{1}{2} \cos 2 \theta$