What is connection here between $x$ and interval: $\sin^{-1}(2x(\sqrt{1-x^2})=2\sin^{-1}x$ .

Question

For the expression

\begin{equation*} \sin^{-1}(2x(\sqrt{1-x^2})=2 \sin^{-1}x,~x\in[\frac{-1}{\sqrt2}, \frac{1}{\sqrt2}], \end{equation*}

I know there is a connection between the interval and the value of $x$, what is that connection and why we are taking a specific value of $x$ in this interval, and why the value of $x$ is changed with interval.

See something more generic : http://math.stackexchange.com/questions/672575/proof-for-the-formula-of-sum-of-arcsine-functions-arcsin-x-arcsin-y — lab bhattacharjee, May 06 '15 at 14:31
Just carefully see for the domains and angle. You'd easily see the two idenities where once it's cos inverse and other is sine inverse. — Someone, May 06 '15 at 14:34

score 2 · Answer 1 · answered May 06 '15 at 14:49

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answered May 06 '15 at 14:49

RE60K

17,716

@ADGif we take $x=cos^{-1}$?,then what'd happen. – Sharma May 06 '15 at 15:07
@Sharma unclear what you want to say? – RE60K May 06 '15 at 15:33
@Sharma why do you want to do that? what is the need of it? it'll make it $|\arccos x|<1\implies |x|>\cos 1$ and maybe some other minor changes too – RE60K May 06 '15 at 15:53
@ADGif we substitue x=cosx in above problem ,then it will define in this interval? – Sharma May 06 '15 at 15:54
@ADGwe take x=sin to prove above situation not cos why? – Sharma May 06 '15 at 15:57
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@Sharma that'll make it a little longer and a little tedious – RE60K May 06 '15 at 16:01

What is connection here between $x$ and interval: $\sin^{-1}(2x(\sqrt{1-x^2})=2\sin^{-1}x$ .

1 Answers1