Solving the equation $3\cos x+\frac{1}{2}=\cos ^{2}x\left(1+2\cos x\left(1+4\sin ^2x\right)\right)$ Solve the equation:$$3\cos x+\frac{1}{2}=\cos ^{2}x\left(1+2\cos x\left(1+4\sin ^2x\right)\right)$$
To solve it, I tried writing the equation in term of $\cos x$. ( I denote $\cos x$ by $c$):
$$3c+\frac12=c^2(1+2c(5-4c^2))$$
$$3c+\frac12=c^2+10c^3-8c^5$$
$$16c^5-20c^3-2c^2+6c+1=0 \qquad\text{Where $c\in[-1,1]$}$$
I tried $c=\pm1,\pm\frac12,0$, but neither of them satisfied the equation. So I don't know how to find $\cos x$.

Your calculations are right. We have $c=\pm\frac 12\sqrt 2$ as two of its roots. Hence, we get $x\in\{\pm\frac{\pi}{4}+k\pi\mid k\in\mathbb Z\}$.
