Pick the Right cos 2x

cos 2x has three forms: ${\cos\left(x\right)}^{2}-{\sin\left(x\right)}^{2}$, $2{\cos\left(x\right)}^{2}-1$ and $1-2{\sin\left(x\right)}^{2}$. Pick the one in the same function as the rest of the equation, so one function is left.

In $\cos\left(2x\right) = \cos\left(x\right)$ the rest is in cosine, so use 2cos²x − 1: $2{\cos\left(x\right)}^{2}-\cos\left(x\right)-1 = 0$. That is a quadratic in cos x, $\left(2\cos\left(x\right)+1\right)\left(\cos\left(x\right)-1\right) = 0$, so cos x = −1/2, giving 120° and 240°, or cos x = 1, giving 0°.