### 5.22.18 Rewrite trigonometric functions as function of tan(x/2) and hyperbolic functions as function of exp(x): halftan_hyp2exp

halftan_hyp2exp takes as argument a trigonometric and hyperbolic expression.
halftan_hyp2exp rewrites sin(x), cos(x), tan(x) in terms of tan(x/2) and sinh(x), cosh(x), tanh(x) in terms of exp(x).
Input :

halftan_hyp2exp(tan(x)+tanh(x))

Output :

(2*tan(x/2))/((1-(tan(x/2))`^`2))+(((exp(x))`^`2-1))/ (((exp(x))`^`2+1))

Input :

halftan_hyp2exp(sin(x)`^`2+cos(x)`^`2-sinh(x)`^`2+cosh(x)`^`2)

Output, after simplification with normal(ans()) :

2