Question

The fundamental period of $\cos \left(\cos 2x\right)+\cos \left(\sin 3x\right)$ is $\frac{\pi }{x}$. Then find $x$.

Answer 1

$\cos \left(\cos 2x\right)+\cos \left(\cos 3x\right)$

$\cos \left(\cos 2(\frac{\pi }{2}+x)\right)+\cos \left({\cos }^2(\frac{\pi }{3}+x)\right)$

$\cos \left(\cos \right(\pi +2x\left)\right)+\cos (\cos \pi +3x)$

$\cos (-\cos (2x\left)\right)+\cos (-\cos 3x)$

$\cos \left(\cos \right(3x\left)\right)+\cos \left(\cos 3x\right)$

$LCM$ of $\frac{\pi }{2},\frac{\pi }{3}=\pi$

$\frac{\pi }{2}=\frac{\pi }{1}$

$x=1$