TITLE: Solve the equation: $\cos^2(x)+\cos^2(2x)+\cos^2(3x)=1$
QUESTION [5 upvotes]: How to solve the equation 

$$\cos^2(x)+\cos^2(2x)+\cos^2(3x)=1$$

Can anyone give me some hints?

REPLY [2 votes]: Using Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$,
$$\cos^2(x)+\cos^2(2x)+\cos^2(3x)-1$$
$$=\cos^2(x)-\sin^23x+\cos^2(2x)$$
$$=\cos(3x-x)\cos(3x+x)+\cos^2(2x)$$
$$=\cos2x(\cos4x+\cos2x)$$
Now use Prosthaphaeresis Formulas on $$\cos4x+\cos2x$$
Should I use a single word more?

Alternatively using $\cos2A=2\cos^2A-1,$
$$\cos^2(x)+\cos^2(2x)+\cos^2(3x)=1$$
$$\iff\cos2x+\cos4x+\cos6x+1=0$$
$$\cos6x=2\cos^23x-1$$ and use Prosthaphaeresis Formula on $$\cos2x+\cos4x$$