Never Divide by sin x

Faced with $2{\cos\left(x\right)}^{2} = \cos\left(x\right)$, it is tempting to divide by cos x: 2 cos x = 1, so 60° and 300°. Both are right, but two answers are gone. Where cos x is 0, at 90° and 270°, both sides are 0 and the equation holds. Dividing by cos x divided by zero there.

It is the trap of ${x}^{2} = 3x$, where dividing by x keeps 3 and loses 0. Move everything to one side instead, $2{\cos\left(x\right)}^{2}-\cos\left(x\right) = 0$, factor, $\cos\left(x\right)\left(2\cos\left(x\right)-1\right) = 0$, and split. The factor you would have divided away becomes its own case.