Hyperbolic Functions

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: