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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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1 Answer 1

up vote 1 down vote accepted

$\sin 2x=2\sin x\cos x$, 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 $$\cos \theta=\pm\sqrt{1-\sin^2\theta}$$

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I facepalmed so hard when I read the first line of your answer. I can't believe I didn't think of that! But anyway, thanks so much. –  Baga Jr. Apr 24 at 17:57
    
lol no problem :) –  NotNotLogical Apr 24 at 18:00

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