Simplifying nested inverse trig function

jakew1198

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Joined
Jul 20, 2018
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1
Hi all!

I came across this specific expression while doing a riddle: arcsin(sin^2 x). Can this be simplified to something more elegant (without using any inverse trig functions)?

I realize arcsin(sin x) = x + 2k(pi), but I'm not sure how to go about more complicated expressions such as this one. In general, are there strategies to go about simplifying expressions like sin(cos(sin(cos x)))?


​Thank you :)
 
Hi all!

I came across this specific expression while doing a riddle: arcsin(sin^2 x). Can this be simplified to something more elegant (without using any inverse trig functions)?

I realize arcsin(sin x) = x + 2k(pi), but I'm not sure how to go about more complicated expressions such as this one. In general, are there strategies to go about simplifying expressions like sin(cos(sin(cos x)))?


​Thank you :)
Let:

\(\displaystyle \displaystyle{sin^{-1}[sin^2(x)] \ = \ \theta + 2k\pi}\)

\(\displaystyle \displaystyle{sin^2(x) \ = \ sin(\theta + 2k\pi)}\)

........
 
Hi all!

I came across this specific expression while doing a riddle: arcsin(sin^2 x). Can this be simplified to something more elegant (without using any inverse trig functions)?

I realize arcsin(sin x) = x + 2k(pi), but I'm not sure how to go about more complicated expressions such as this one. In general, are there strategies to go about simplifying expressions like sin(cos(sin(cos x)))?


​Thank you :)
1st of all sin(cos(sin(cos x))) is NOT an example of nested inverse function.

Let sin(cos(sin(cos x))) = y. Then x = cos-1(sin-1(cos-1(sin-1(y)))). Do you see how I got this?
 
Hi all!

I came across this specific expression while doing a riddle: arcsin(sin^2 x). Can this be simplified to something more elegant (without using any inverse trig functions)?

I realize arcsin(sin x) = x + 2k(pi), but I'm not sure how to go about more complicated expressions such as this one. In general, are there strategies to go about simplifying expressions like sin(cos(sin(cos x)))?

My immediate reaction was that it seems unlikely that you can rewrite arcsin(sin^2 x) as an algebraic function of x, which I think is what you want. You might consider the power-reduction identities, but I don't think that will get you where you want to be.

What you want to do is a very special circumstance, not something to expect in most cases. Certainly there is no way to simplify trig function compositions like sin(cos(sin(cos x))), or even just sin(cos(x)).

Does the context of your question lead you to think you should be able to simplify this, or is it just your own curiosity?
 
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