How to prove $\tanh^{-1}(\sin x)=\sin^{-1}(\tan x)$

Question

Here's what I attempted:

$$ y =\tanh^{-1}(\sin x)$$

$$\tanh y=\sin x$$ But I don't know what to do after this. Please help me.

Answer 1

These two are not equal. Let $x = \pi/3$ for instance. Then

$$\text{arctanh}(\sin(x)) \approx 1.32$$

(per Wolfram), while you can't even get a real-number result for the latter equation. This is because

$$\arcsin(\tan(\pi/3)) = \arcsin( \sqrt 3)$$

but the domain for arcsine (for real outputs) is $[-1,1]$. In particular, Wolfram approximates the answer as about $1.57 - 1.15i$.