Use Identities to Solve an Equation

Use identities to solve exactly the trigonometric equation over the interval $0 \leq x < 2 π .$

\cos x \cos (2 x) + \sin x \sin (2 x) = \frac{\sqrt{3}}{2}

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
Notice that the left side of the equation is the difference formula for cosine.

$\begin{matrix} \cos x \cos (2 x) + \sin x \sin (2 x) & = & \frac{\sqrt{3}}{2} \\ \cos (x - 2 x) & = & \frac{\sqrt{3}}{2} & Difference formula for cosine \\ \cos (- x) & = & \frac{\sqrt{3}}{2} & Use the negative angle identity . \\ \cos x & = & \frac{\sqrt{3}}{2} \end{matrix}$
From the unit circle in Figure 2, we see that $\cos x = \frac{\sqrt{3}}{2}$ when $x = \frac{π}{6}, \frac{11 π}{6} .$

How did it go?