Hyperbolic functions simplifying

Question

How do you simplify

$$\cosh(\sinh^{-1}(x))$$

to become

$$(1+x^2)^{1/2}$$

I have managed to get $(1+\sinh^2(\sinh^{-1}(x))^{1/2}$ but haven't been able to progress from there.

Answer 1

From $$\cosh^2t-\sinh^2t=1,$$ draw

$$\cosh t=\sqrt{1+\sinh^2t}$$ and set $$t=\sinh^{-1}x.$$

Answer 2

By definition you have $$cosh(x)=\frac{e^x+e^{-x}}{2}$$ and $$ sinh(x)=\frac{e^x-e^{-x}}{2}$$ You can compute the inverse function and find $$sinh^{-1}(x)=ln(x+\sqrt{x^2+1})$$ Which substituted in your expression gives $$\frac{ e^{ln{(x+\sqrt{x^2+1})}}+e^{-ln(x+\sqrt{x^2+1})}}{2}$$

Now remember that logarithm is the inverse of exponential, notice that $\frac{1}{a+b}=\frac{1}{a+b}\cdot \frac{a-b}{a-b}$ and $-log(x)=log(\frac{1}{x})$. Your result will follow.

Answer 3

Since Tavish has already given the crucial key in a comment, I just add another way using the differentiation rule for inverse functions:

Set $$f(x) = \cosh(\sinh^{-1}x)$$

Then $$f'(x) = \frac x{f(x)} \Rightarrow \left([f(x)]^2\right)' = 2x$$

Hence,

$$[f(x)]^2 = x^2+c \stackrel{f>0, f(0)=1}{\Longrightarrow}f(x) =\sqrt{x^2+1}$$