Question

Find the general solution for the following equation:

cos 3x+ cos x - cos 2x = 0

Answer 1

cos 3x + cos x - cos 2x = 0

$\Rightarrow$ 2 cos $\left(\frac{3x+x}{2}\right)cos\left(\frac{3x-x}{2}\right)$ - cos 2x = 0

$\Rightarrow$ 2 cos 2x cos x - cos 2x =0

$\Rightarrow$ cos 2x (2 cos x-1) = 0

Either cos 2x = 0 or 2 cos x - 1 = 0

$\Rightarrow$ 2x = (2n+1) $\frac{\pi }{2}$

or cos x = $\frac{1}{2}$ = cos $\frac{\pi }{3}$, n $ϵ$ Z

$\Rightarrow$ x = (2n+1) $\frac{\pi }{4}$

or x = $2n\pi \pm \frac{\pi }{3}$, n $ϵ$ Z.