how many $x \in {[{-\pi},\pi]}$ such that

$\cos 3x + \cos x = 0$

The answer is 6.

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cos3x = 4cos^3(x) - 3cosx

so 4cos^3(x) = 2cosx

hence cosx = 0 or cos^2(x) = 1/2

the former has two solution viz pi/2 and -pi/2.

cos^(2)x is periodic with pi .

Two solutions in (0,pi) hence total of 4 solutions.

Finally adding a grand of 6 solutions