I remember reading somewhere that we can simplify $\cos(\arcsin x)$ and $\sin(\arccos x)$ in terms of a polynomial by making the substitution $m=\arcsin x$ or $m=\arccos x$ (respectively), then constructing a right angle triangle with appropriate ratios. Does anyone know what I'm talking about? I don't really remember how to do this, so if someone can show me I would really appreciate it!
Thanks.
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$\begingroup$$$\cos(\arcsin x) = \sqrt{1 - \sin(\arcsin x)^2} = \sqrt{1-x^2}$$
This is the quick and dirty solution, I'll leave you to figure out the intricacies of the sign and the respective domains. Of course, you can simplify the other expression the same way.
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