<s>[INST] Solve $\cos3x - 18\cos x +10 =0$ I want to solve
$$\cos 3x - 18\cos x +10 =0 $$
I tried:
1) Replacing $\cos 3x$ to $\cos^3x - 3\cos x$
2) Replacing $\cos x$ to $t$ we get:
$$t^3 - 21t +10 = 0$$
So we get cubic equation. But I can't solve it.
 [/INST] Given $$cos3x - 18cosx +10 =0 $$ $$\implies 4\cos^3x-3\cos x-12\cos x+10 = 0$$ Factorizing the cubic equation as follows  $$\left(\cos x-\frac{1}{2}\right) \left(\cos x-2\right)\left(\cos x+\frac{5}{2}\right)= 0$$ $$\text{if}\ \cos x-\frac{1}{2}=0 \implies \cos x=\frac{1}{2}\iff x=2n\pi\pm\frac{\pi}{3} $$ $$\text{if}\ \cos x-2=0 \implies \cos x\neq 2$$$$\text{if}\ \cos x+\frac{5}{2}=0 \implies \cos x\neq \frac{-5}{2}$$  Hence, the general solution for $x$ is $$x=2n\pi\pm \frac{\pi}{3}$$ Where, $n$ is any integer
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