Solving the Equation Using a Double-Angle Formula

Solve the equation exactly using a double-angle formula: $\cos (2 θ) = \cos θ .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We have three choices of expressions to substitute for the double-angle of cosine. As it is simpler to solve for one trigonometric function at a time, we will choose the double-angle identity involving only cosine:

$\begin{matrix} \cos (2 θ) & = & \cos θ \\ 2 \cos^{2} θ - 1 & = & \cos θ \\ 2 \cos^{2} θ - \cos θ - 1 & = & 0 \\ (2 \cos θ + 1) (\cos θ - 1) & = & 0 \\ 2 \cos θ + 1 & = & 0 \\ \cos θ & = & - \frac{1}{2} \\ \cos θ - 1 & = & 0 \\ \cos θ & = & 1 \end{matrix}$
So, if $\cos θ = - \frac{1}{2},$ then $θ = \frac{2 π}{3} \pm 2 π k$ and $θ = \frac{4 π}{3} \pm 2 π k;$ if $\cos θ = 1,$ then $θ = 0 \pm 2 π k .$

How did it go?