<s>[INST] Solve : $2\sin^3 (x) = \cos (x)$ 
Solve : $2\sin^3 (x) = \cos (x)$

How to solve the above equation?
I tried, but failed to succeed. 
Graph of the following equation
But how to solve the equation without graph?
 [/INST] Here's how I solved it by hand:
Notice that if we square both sides, we get $$4 \sin^6 x = \cos^2 x$$
Using the Pythagorean identity,
$$4 \sin^6 x + \sin^2 x - 1 = 0$$
Let $y = \sin^2 x$. Our equation becomes $$4y^3 + y - 1 = 0$$ Normally, we would have to use the cubic equation formula, but according to the Rational Root Theorem, we can brute force our way through the eligible rational roots: $\pm 1$,$\pm \frac{1}{2}$, and $\pm \frac{1}{4}$. We notice that $\frac{1}{2}$ is indeed a solution. Dividing out, we see that the remaining roots are roots of $2y^2 + y + 1$, which are complex. Thus, all we have to do is find the numbers for which $$\sin^2 x = \frac{1}{2}$$ You should be able to get that $\pm \frac{\pi}{4}$ and $\pm \frac{5\pi}{4}$ are solutions. We see $-\frac{\pi}{4}$ and $-\frac{5\pi}{4}$ are extraneous solutions. Thus, our solution set is all $\frac{\pi}{4} + k\pi, k\in \mathbb{Z}$
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