# help with hyperbolic functions like sinh and tanh

Show that $\sin^{-1}(\tanh x)=\tan^{-1}(\sinh x)$. Got a hint that $\sin\theta=\tanh x$ but I still don't know how to proceed...

• Does $\tan hx$ denote $\tan (hx)$ or $(\tan h) \times x$? – Paul Oct 3 '14 at 2:07
• @Paul: Neither. $\tanh x$ denotes the hyperbolic tangent of x. – Lucian Oct 3 '14 at 3:13
• @Lucian Thanks. I see! – Paul Oct 3 '14 at 3:49

Differentiate both to show they share the same derivative. Then, check one value to show equality is met. (if $f'(x)=g'(x)$ on a connected domain then $f(x)=g(x)+c$. You want $c=0$)
Let $\sin\theta=\tanh x$. $\sin^{-1}(\tanh x)=\sin^{-1}(\sin\theta)=\theta$.
We only need to prove $$\sinh x= \tan \theta.$$ To prove it, by using $\sin\theta=\tanh x$, it only need to prove that
$$\cos \theta =\frac{1} {\cosh x}.$$
In fact: $$\cos^2 \theta=1-\sin^2 \theta=(1-\sin \theta)(1+\sin \theta)=\frac{4}{(e^x+e^{-x})^2}.$$