This problem was found on an old post and i thought it was an interesting problem. a hint was given which said remember that tan x = cot(pi/2 - x). i tried letting u = pi/2 - x and i got:
i'm not sure what to do next though. my substitution didn't seem to do much.
i'm having some trouble with the first integral you posted. i tried turning sin(x) into cos(x)/cot(x) and then using the substitution x = pi/2 - u, i then added the two integrals together and divided by 2. but i couldn't simplify the integral like shown above for the first integral i posted. i also tried making some u substitutions first but all they did was make the integral more complicated. how should this integral be tackled?