# Trig and Inverse Trig Function Compositions

Sorry if this is a dumb question, but I honestly tried searching and all I could find was obvious stuff like $\sin(\arcsin(x)) = x$

So what is the logic behind simplifying expressions like this, where there is a constant or something else along with the $\arcsin(x)$?

$\sin(2\arcsin(x))=2x\sqrt{1-x^2}$

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$\sin 2x=2\sin x\cos x$, and also $$\cos \arcsin x=\sqrt{1-\sin^2(\arcsin x)}=\sqrt{1-x^2}$$ Therefore $$\sin 2\arcsin x=2\sin \arcsin x\cos\arcsin x=2x\sqrt{1-x^2}$$
However, make sure to check that the signs are correct. Technically $$\cos \theta=\pm\sqrt{1-\sin^2\theta}$$