Introduction to Hyperbolic Functions

Prove that $\frac{\cosh x}{1-\tanh x}+\frac{\sinh x}{1-cothx}=\sinh x+\cosh x$, for $x\ne 0$.

Prove that $\frac{\tanh x}{sechx-1}+\frac{\tanh x}{sechx+1}=-2cosechx$ for $x\ne 0$

Prove that $(\cosh x+\sinh x{)}^n=\cosh (nx)+\sinh (nx)$ for any $n\in \mathrm{R}$.

Prove that $(\cosh x-\sinh x{)}^n=\cosh (nx)-\sinh (nx)$ for any $n\in \mathrm{R}$.

If $\sinh x=\frac{3}{4},$ then find $\cosh \left(2x\right)$ and $\sinh \left(2x\right)$.