<s>[INST] Number of solutions of $8\sin(x)=\frac{\sqrt{3}}{\cos(x)}+\frac{1}{\sin(x)}$ Number of solution in $[0,2\pi]$ satisfying the equation 
$$8\sin(x)=\frac{\sqrt{3}}{\cos(x)}+\frac{1}{\sin(x)}$$
Options are $5$ or $6$ or $7$ or $8$.
Doing some manipulations I reached $4\sin^2(x)\cos(x)=\sin(x+\pi/6)$ but I think this direction will not lead towards result. How to proceed?
 [/INST] Multiply through by $\sin x\cos x$. We get 
$$8\cos x(1-\cos^2 x)=\sqrt{3}\sin x+\cos x.$$
Recall that $\cos 3x=4\cos^3 x-3\cos x$. So the left-hand side becomes $8\cos x-2(\cos 3x+3\cos x)$, that is, $2\cos x-2\cos 3x$. Thus our equation can be rewritten as
$$\cos3x=\frac{1}{2}\cos x-\frac{\sqrt{3}}{2}\sin x.$$
The right-hand side is equal to $\cos(x+\pi/3)$, so our equation simplifies to
$$\cos 3x=\cos(x+\pi/3).$$
To finish, recall that $\cos s=\cos t$ if and only if $s=\pm t+2k\pi$ for some integer $k$.
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