Question: Integrate sec(x)/(sec(x)+tan(x) )
sec(x)/(sec(x)+tan(x) )=1/(1+sin(x) )
1/(1+sin(x) )=1/(1+2∙sin(x/2)∙cos(x/2) )
=sec^2(x/2)/(sec^2(x/2)+2∙tan(x/2) )
=sec^2(x/2)/(1+tan^2(x/2)+2∙tan(x/2) )
=sec^2(x/2)/(1+tan(x/2) )^2
Now it is a simple integration.
∫sec^2(x/2)/(1+tan(x/2) )^2 dx=(-2)/(1+tan(x/2) )
Answer: ∫sec(x)/(sec(x)+tan(x) ) dx = (-2)/(1+tan(x/2) )