Question

Find the general solution of the equation $\cos 3x+\cos x-\cos 2x=0$

Answer 1

Step 1: Simplication
Given: $\cos 3x+\cos x-\cos 2x=0$
$\Rightarrow (\cos 3x+\cos x)-\cos 2x=0$
$\Rightarrow 2\cos \left(\frac{3x+x}{2}\right)\cos \left(\frac{3x-x}{2}\right)-\cos 2x=0$
$\Rightarrow 2\cos \left(\frac{4x}{2}\right)\cos \left(\frac{2x}{2}\right)-\cos 2x=0$
$\Rightarrow 2\cos 2x\cos x-\cos 2x=0$
$\cos 2x=0$ or $2\cos x-1=0$
$\cos 2x=0$ or $\cos x=\frac{1}{2}$

Step $2:$ General solution for $\cos 2x=0$
The general solution is
$2x=(2n+1)\frac{\pi }{2}$
$x=(2n+1)\frac{\pi }{4}$ where $n\in Z$

Step $3:$ General solution for $\cos x=\frac{1}{2}$
$\Rightarrow \cos x=\cos \frac{\pi }{3}$
We know that general solution for $\cos x=\cos y$ is $x=2n\pi \pm y,n\in Z$

Put $y=\frac{\pi }{3}$
$x=2n\pi \pm \frac{\pi }{3},$ where $n\in Z$