### Hyperbolic Functions

goals
If $'x'$ is an acute angle and $y = \log_e\left(\tan\,\left(\dfrac{\pi }{4} + \dfrac{x}{2}\right)\right)$ then $\cos x.\cos y=$...........
For $\quad z = x + iy, \space x, y \in R$

Define $\quad e^z = e^x(\cos y + i\sin y)$ ,$\quad \sin h z = \displaystyle\frac{1}{2}(e^z - e^{-z})$ ,$\quad \cos h z = \displaystyle\frac{1}{2}(e^z + e^{-z})$
$\displaystyle \sec h^{-1}(\frac{1}{5})=$
Let $y=\tan { { h }^{ -1 }x }$ so $\tan { hy } =x$
Express $\tan { hy }$ in terms of ${e}^{y}$ and hence show that
$\quad { e }^{ 2y }=\cfrac { 1+x }{ 1-x }$
Deduce that $\tan { { h }^{ -1 }x } =\cfrac { 1 }{ 2 } \ln { \left( \cfrac { 1+x }{ 1-x } \right) }$
(Do not forget that $\tan { { h }^{ -1 }x }$ is only defined for $\left| x \right| < 1$)
The graph of the hyperbolic sine function for all real values is: