Rewrite sin, cos, tan in terms of tan(x/2) : halftan

halftan takes as argument an expression containing trigonometric functions.
halftan rewrites sin(x), cos(x) and tan(x) in terms of tan(${\frac{{x}}{{2}}}$).
Input :

halftan(sin(2*x)/(1+cos(2*x)))

Output :

2*tan(2*x/2)/((tan(2*x/2))^2+1)/

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

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

tan(x)

Input :

halftan(sin(x)^2+cos(x)^2)

Output :

(2*tan(x/2)/((tan(x/2))^2+1))^2+

((1-(tan(x/2))^2)/((tan(x/2))^2+1))^2

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

1