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Why is it that $\arctan \theta \neq \dfrac{\arcsin\theta}{\arccos\theta}$ ?

Thanks for your help in figuring this out.

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2 Answers

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For one thing, the principal value of arctan is from $0$ to $\pi$ on Monday, Wednesday, and Friday, and from $-\pi/2$ to $\pi/2$ on Tuesday, Thursday, and Saturday.

However $\frac{\arcsin x}{\arccos x}$ is unbounded as $x \to \pi/2$, so this can not be a value of $\arctan$.

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I hope the quite elementary figure below can help you.

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