Question

The number of solutions of the equation
$1+sin^{4}x=cos^{2}\,3x, x\,\in\left[-\frac{5\pi}{2}, \frac{5\pi}{2}\right]$ is

Answer 1

Consider equation, $1+sin^{4}x = cos^{2}3x$
$L.H.S. =1+sin^{4}\,x$ and $R.H.S. = cos^{2} 3x$
$ ⋌ L.H.S. \ge\,1$ and $R.H.S. \le\,1 $
$\,\Rightarrow \,L.H.S.=R.H.S.=1$
$sin^{4}x=0$, and $cos^{2}\,3x=1$
$\Rightarrow \,sin\,x=0$ and $(4\,cos^{2}x-3)^{2}\,cos^{2}x=1$
$\Rightarrow \,sin\,x=0$ and $cos^{2}x=1$
$\Rightarrow x=0, \pm\pi, \pm2 \pi$
Hence, total number of solutions is 5