Question

$\sqrt{3sin(5x)-cos(x)^{2}-3 } = 1-sin(x)$

Answer 1

$\sqrt{3sin5x-cos^2x-3}=1-sinx\\3sin5x-cos^2x-3=1-2sinx+sin^2x,1-sinx\geq 0\\ 3sin5x-3=2-2sinx\Leftrightarrow 3sin5x+2sinx=5\\48sin^5x-60sin^3x+17sinx-5=0\\(sinx-1)\left ( 48sin^4x+48sin^3x-12sin^2x-12sinx+5 \right )=0\\(sinx-1)\left (\left ( 2sinx-\frac{5}{4}\right )^2+\frac{2}{3} \right )=0\\sinx=1\Rightarrow x=\frac{\pi}{2}+2\pi k,k\in \mathbb{Z}$