If $\sin x(\cos \frac{x}{4}-2\sin x)+(1+\sin \frac{x}{4}-2\cos x)\cos x=0$, then $x =$

$\sin \frac{5x}{4}+\cos x=2\dots \text{(i)}$ Now equation(i) will hold if

$\frac{5x}{4}=2n\pi +\frac{\pi }{2},n\in Z=\frac{(8n+2)\pi }{5},n\in Z....\left(ii\right)$

and $x=2m\pi ,m\in Z.....\left(iii\right)$

therefore the general solution of the given equation can be obtained by substituting either $m = 4k–3$ in equation(iii) or $n = 5 p–4$ in equation (ii). therefore the general solution of equation (i) is $x=(8k-6)\pi ,k\in Z$