Prove that `int(g(sinx))/(g(sinx)+g(cosx))dx = int_(0)^(pi/2)(g(cosx))/(g(sinx)+g(cosx))dx = (pi)/(4)`.

Explanation

Let `=overset(pi/2)underset(0)int(g(sinx))/(g(sinx)+g(cosx))dx rArr I = underset(0)overset(pi/2)int(g(sin(pi/2-x)))/(g(sin(pi/2-x))+g(cos(pi/2-x)))dx`
`=overset(pi/2)underset(0)int(g(cosx))/(g(cosx)+g(sinx))dx`
on adding, we obtain
` = 2I = overset(pi/2)underset(0)int((g(sinx))/(g(sinx)+g(cosx))+(g(cosx))/(g(cosx)+g(sinx)))dx =overset(pi/2)underset(0)int dx rArr I = (pi)/(4)`.