TITLE: What is the general solution to $2 \cos^2 x-\cos x=0$?
QUESTION [6 upvotes]: What is the general solution to this trig equation?
$$2 \cos^2 x-\cos x=0$$
Thanks.

REPLY [19 votes]: To make it easier to visualise and solve, you can substitute $u=\cos{x}$.
$$2\cos^2{x}-\cos{x}=0$$
$$2u^2-u=0$$
From here, you can either use the quadratic formula or factorise. I chose to factorise:
$$u(2u-1)=0$$
Hence we have the solutions $u=0,\frac{1}{2}$.
Now, we solve the equations resulting from our substitution $u=\cos{x}$:
$$\cos{x}=0 \tag{1}$$
$$\cos{x}=\frac{1}{2} \tag{2}$$
For equation $(1)$, we obtain:`
$$x=\frac{\pi}{2}+k\pi$$
For equation $(2)$, we obtain:
$$x=2k\pi+\frac{\pi}{3}$$
And
$$x=2k\pi-\frac{\pi}{3}$$
Where $k \in \mathbb{Z}$.
Hence, our general solution is:
$$x=\begin{cases} \frac{\pi}{2}+k\pi \\ 2k\pi+\frac{\pi}{3} \\ 2k\pi-\frac{\pi}{3} \end{cases}$$