Help and explaination please!

This should help:

Use the identity

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

Then,

$\displaystyle \cos(12x)+\cos(16x)=2\cos(14x)\cos(2x)\\ =0\\ \text{ when } \\ 14x = \pi/2 + m\pi \\ \text{or when } \\ 2x = \pi/2 +n\pi, \ \text{ m and n integers.}$

Smallest will be when m = 0, then x = pi/28, etc.