If `int_0^(pi/2)logsinthetadtheta=k ,` then find the value of `int_pi^(pi/2)(theta/(s intheta))^2dtheta` in terms of `k`

Explanation

`I=int_(0)^(pi//2)((theta)/(sin theta))^(2)d theta`
`=int_(0)^(pi//2) theta^(2) cosec^(2) theta d theta`
`=[theta^(2) (-cot theta)]_(0)^(pi//2)-int_(0)^(pi//2) 2theta (-cot theta) d theta`
(Integrating by parts)
`=[lim_(theta to oo) theta^(2).cot theta]+2 int_(0)^(pi//2) theta cot theta d theta`
`=0+2[[ theta log sin theta]_(0)^(pi//2)-int_(0)^(pi//2) log sin theta d theta]`
(Integrating by parts)
`=2[-lim_(theta to oo) theta n sin theta -k]`
`=-2k`