Why does $\frac{1}{\sin x} = 2\sin x$? 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?

$$\frac1{\frac{\sqrt2}2}=\sqrt2.
$$
