Trigonometry addition formula sinh(x - y) = sinh x cosh y - cosh x sinh y 8080624

We will show that for any real element x, y the trigonometric formula (difference identity) sinh(x - y) = sinh x cosh y - cosh x sinh y

Difficult Proof/Demonstration

We start from the left hand side of the equality:

$$\begin{array}{rl}\sinh (x-y)& =\frac{e^{x+y}-e^{-(x-y)}}{2}\\ & =\frac{e^{x+y}-e^{-x+y}}{2}\\ & =\frac{2e^{x-y}-2e^{-x+y}}{4}\\ & =\frac{2e^{x-y}-2e^{-x+y}+(e^{x+y}-e^{-x-y})-(e^{x+y}-e^{-x-y})}{4}\\ & =\frac{2e^{x-y}+(e^{x+y}-e^{-x-y})-(e^{x+y}-e^{-x-y})-2e^{-x+y}}{4}\\ & =\frac{e^{x-y}+e^{x-y}+(e^{x+y}-e^{-x-y})-(e^{x+y}-e^{-x-y})-e^{-x+y}-e^{-x+y}}{4}\\ & =\frac{e^{x-y}-e^{-x+y}+(e^{x+y}-e^{-x-y})+e^{x-y}-(e^{x+y}-e^{-x-y})-e^{-x+y}}{4}\\ & =\left(\frac{e^{x-y}+e^{x+y}-e^{-x-y}-e^{-x+y}}{4}\right)+\left(\frac{e^{x-y}-e^{x+y}+e^{-x-y}-e^{-x+y}}{4}\right)\\ & =\left(\frac{e^x-e^{-x}}{2}\right)\left(\frac{e^y+e^{-y}}{2}\right)+\left(\frac{e^x+e^{-x}}{2}\right)\left(\frac{e^{-y}-e^y}{2}\right)\\ & =\left(\frac{e^x-e^{-x}}{2}\right)\left(\frac{e^y+e^{-y}}{2}\right)-\left(\frac{e^x+e^{-x}}{2}\right)\left(\frac{e^y-e^{-y}}{2}\right)\\ & =\sinh x\cosh y-\cosh x\sinh y\end{array}$$

Easy Proof/Demonstration

We start from the right hand side of the equality:

$$\begin{array}{rl}\sinh x\cosh y-\cosh x\sinh y& =\left(\frac{e^x-e^{-x}}{2}\right)\left(\frac{e^y+e^{-y}}{2}\right)-\left(\frac{e^x+e^{-x}}{2}\right)\left(\frac{e^y-e^{-y}}{2}\right)\\ & =\left(\frac{e^{x+y}+e^{x-y}-e^{-x+y}-e^{-x-y}}{4}\right)-\left(\frac{e^{x+y}-e^{x-y}+e^{-x+y}-e^{-x-y}}{4}\right)\\ & =\frac{2e^{x-y}-2e^{-x+y}}{4}\\ & =\frac{e^{x+y}-e^{-x+y}}{2}\\ & =\frac{e^{x+y}-e^{-(x-y)}}{2}\\ & =\sinh (x-y)\end{array}$$