How to evaluate hyperbolic functions, involving inverses, by hand?

Question

How does one evaluate $$\sinh(2{\sinh^{-1}{(2)}})$$ by hand?

Answer 1

HINT

We have

$$\sinh(2x)=2\sinh x \cosh x$$

then use that

• $ \sinh(\operatorname{arsinh}x) = x $

• $ \cosh(\operatorname{arsinh}x) = \sqrt{1+x^{2}} $