Answer
$$x=2\pi n,\:x=\pi +2\pi n,\:x=\frac{5\pi }{4}+2\pi n,\:x=\frac{7\pi }{4}+2\pi n,\:x=\frac{\pi }{4}+2\pi n,\:x=\frac{3\pi }{4}+2\pi n$$
Work Step by Step
We solve the equation using the properties of trigonometric functions. Note, there is a general solution since trigonometric identities go up and down and this can pass through a given value of y many times. Solving this, we find:
$$2\sin ^3\left(x\right)-\sin \left(x\right)=0 \\ \sin \left(x\right)=0,\:\sin \left(x\right)=-\frac{\sqrt{2}}{2},\:\sin \left(x\right)=\frac{\sqrt{2}}{2} \\ x=2\pi n,\:x=\pi +2\pi n,\:x=\frac{5\pi }{4}+2\pi n,\:x=\frac{7\pi }{4}+2\pi n,\:x=\frac{\pi }{4}+2\pi n,\:x=\frac{3\pi }{4}+2\pi n$$
