# Question #fce84

Jun 8, 2016

$x = \frac{\pi}{2} + k \pi , \frac{\pi}{4} + \frac{k \pi}{2} , k \in \mathbb{Z}$

#### Explanation:

Use the identity $\sin 2 x = 2 \sin x \cos x$ to rewrite the expression.

$\sin 2 x \sin x = \cos x \text{ "=>" } 2 \sin x \cos x \left(\sin x\right) = \cos x$

$\implies \text{ } 2 {\sin}^{2} x \cos x = \cos x$

Subtract $\cos x$ from each side.

$2 {\sin}^{2} x \cos x - \cos x = 0$

Factor $\cos x$.

$\cos x \left(2 {\sin}^{2} x - 1\right) = 0$

Set both of these terms equal to $0$.

$\cos x = 0$

This occurs at $x = \frac{\pi}{2}$ and $x = \frac{3 \pi}{2}$ on the interval $\left[0 , 2 \pi\right)$, but can be generalized as $x = \frac{\pi}{2} + k \pi$ where $k$ is an integer.

The other term is

$2 {\sin}^{2} x - 1 = 0$

${\sin}^{2} x = \frac{1}{2}$

$\sin x = \pm \sqrt{\frac{1}{2}} = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2}$

Sine equals $\pm \frac{\sqrt{2}}{2}$ at $x = \frac{\pi}{4} , \frac{3 \pi}{4} , \frac{5 \pi}{4} , \frac{7 \pi}{4}$ on $\left[0 , 2 \pi\right)$, which can be generalized to $\frac{\pi}{4} + \frac{k \pi}{2}$, where $k$ is an integer.

Note that the mathematical way to express that $k$ is an integer is to say that $k \in \mathbb{Z}$.