How would I solve cos x + cos 2x = 0? Please show steps.

We know that

$\cos 2 x = {\cos}^{2} x - {\sin}^{2} x = {\cos}^{2} x - \left(1 - {\cos}^{2} x\right) = 2 {\cos}^{2} x - 1$

hence the equation is

$\cos x + \cos 2 x = \cos x + 2 {\cos}^{2} x - 1$

Hence we have to solve the

$2 {\cos}^{2} x + \cos x - 1 = 0 \implies \left(\cos x + 1\right) \cdot \left(2 \cos x - 1\right) = 0$

or

$\cos x = - 1 \implies \cos x = \cos \pi \implies x = 2 \cdot k \cdot \pi \pm \pi$

and

$2 \cos x - 1 = 0 \implies \cos x = \frac{1}{2} \implies \cos x = \cos \left(\frac{\pi}{3}\right) \implies x = 2 \cdot k \cdot \pi \pm \frac{\pi}{3}$