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I know how to rewrite trig functions of inverse trig functions, but this is pretty weird to me and I know it doesn't work the same way. I've tried seeing it as $\tan^{-1} x=f(\cos^{-1} x)$ for help, and writing it as the solution to a differential equation, as well as some simple substitutions, however nothing has worked. What are the simplest ways of rewriting this and are there any that don't involve infinite series?

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  • $\begingroup$ Where does this problem come from? It might be possible to simplify it at a step before this one. $\endgroup$
    – Andrei
    May 1, 2019 at 15:36
  • $\begingroup$ It isn't from anything, but I stumbled across it trying to prove that arctan equals arcsin/arccos leads to a contradiction. $\endgroup$ May 1, 2019 at 15:41
  • $\begingroup$ I don't think there is an easier form to write this without using a Taylor series or complex exponentials. $\endgroup$
    – log_math
    May 1, 2019 at 15:53

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There is no real simplification possible for $\arctan(\cos(x))$. Well, you could write it as $$ \arcsin\left(\frac{\cos(x)}{\sqrt{1+\cos^2(x)}}\right)$$

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