So this is the problem:
$$\sin2x=\sqrt2\cos x$$
$$\cos x(2\sin x-\sqrt2)=0$$ My question is, how can I come with this to the conclusion that, according to my answer sheet, $\cos x=0$ and $\sin x=\frac{\sqrt2}{2}$ Is it just plugging in something until it works?
Recall in Algebra you solved problems like $(x-2)(x+3)=0$? You used the Zero Product Property, which @G.Sassatelli stated. So for my example you get $x-2=0$ or $x+3=0$, and solve.
For your equation, we have $(\cos x)\left(2\sin x-\sqrt{2}\right)=0$. We use the Zero Product Property to get $\cos x=0$ or $2\sin x-\sqrt{2}=0$. This second equation, upon solving for $\sin x$, results in $\sin x=\sqrt{2}/2$.