Telangana Board Solutions for Exercise 1: Exercise 9 (a)

Prove that $\tanh \left(x-y\right)=\frac{\tanh x-\tanh y}{1-\tanh x\tanh y}$.

Prove that $coth\left(x-y\right)=\frac{cothx·cothy-1}{cothy-cothx}$.

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 $n\in \mathrm{R}$.

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

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

For any $x\in R$, prove that ${\cosh }^{4}x-{\sinh }^{4}x=\cosh \left(2x\right)$.

If $u={\log }_e\left(\tan \left(\frac{\pi }{4}+\frac{\theta }{2}\right)\right)$ and if $\cos \theta >0$, then prove that $\cosh u=\sec \theta$.