# What can you deduce from cosX(A−B)=0? So this is the problem: sin2x=sqrt(2)cos x cosx(2sinx−sqrt(2)=0 My question is, how can I come with this to the conclusion that, according to my answer sheet, cosx=0 and sinx=sqrt(2)/2 Is it just plugging in something until it works?

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