Problem with hyperbolic functions demostrations

[Rono]

Homework Statement

Prove that $cosh (\frac{x}{2}) = \sqrt{\frac{cosh(x)+1}{2}}$

Homework Equations

$cosh(x) = \frac{e^{x}+e^{-x}}{2}$

The Attempt at a Solution

$\frac{\sqrt{e^{x}}+\sqrt{e^{-x}}}{2} \ast \frac{\sqrt{e^{x}}-\sqrt{e^{-x}}}{\sqrt{e^{x}}-\sqrt{e^{-x}}} \rightarrow \frac{e^{x}+e^{-x}}{2} \ast \frac{1}{\sqrt{e^{x}}-\sqrt{e^{-x}}} \rightarrow cosh(x) \ast \frac{1}{\sqrt{e^{x}}-\sqrt{e^{-x}}}$

After that, I don't know what to do. Would be glad if somebody would tell me what I'm doing wrong or how to do it. Thanks.

 

[kevinferreira]

$$cosh^2\frac{x}{2}=\frac{e^x+e^{-x}+2}{4}=\frac{cosh(x)+1}{2}$$
That's it!