The number of solutions of the equation (x+3)(3-x) = 14^22x, x[-3,3] is:

Question

The number of solutions of the equation $\cos\left(x+\dfrac\pi3\right)\cos\left(\dfrac\pi3-x\right) = \dfrac14\cos^22x$, $x\in[-3\pi,3\pi]$ is:

Accepted Answer

Using $\cos(P+Q)\cos(P-Q) = \cos^2P-\sin^2Q$ with $P=\tfrac\pi3, Q=x$:

$$\cos\left(\tfrac\pi3+x\right)\cos\left(\tfrac\pi3-x\right) = \cos^2\tfrac\pi3 - \sin^2x = \tfrac14-\sin^2x$$

$$\tfrac14-\sin^2x = \tfrac14\cos^22x = \tfrac14(1-\sin^22x) = \tfrac14-\tfrac14\sin^22x$$

$$\sin^2x = \tfrac14\sin^22x = \sin^2x\cos^2x \implies \sin^2x(1-\cos^2x)=0 \implies \sin^4x=0 \implies \sin x=0$$

So $x=n\pi$. Within $[-3\pi,3\pi]$: $x=-3\pi,-2\pi,-\pi,0,\pi,2\pi,3\pi$ --- 7 solutions.

Solutions · 1