algebra precalculus - Why does $frac{1}{sin x} = 2sin x$?

I'm trying to understand the solution of a trigonometry problem. One of the steps of the solution says that:

$$\frac{\sqrt2}{2} = \sin x$$

And then directly deduces that:

$$\sqrt2 = \frac{1}{\sin x}$$

I wonder how this equivalence works. It looks like they multiply both sides of the equation by 2. When I check with a calculator, $\frac{1}{\sin x}$ is indeed equal to $2 \sin x$ for the value of $x$ used in the exercise, which happens to be $\frac{\pi}{4}$, but it doesn't seem to be the case of other values of $x$. What am I missing here?

Answer

Recall that $2=\sqrt 2\cdot\sqrt 2$ and therefore:

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

Now multiply by $\frac{\sqrt 2}{\sin x}$ both sides and you have as needed.