The value of ∫0α dx1-cos⁡αcos⁡x(0<α<π/2) is

Putting tan⁡x2=t the given integral reduces to=2∫0tan⁡α2 dt(1−cos⁡α)+t2(1+cos⁡α)=1cos2⁡α2∫0tan⁡α/2 dtt2+tan2⁡α2=1cos2⁡α2tan⁡α2tan−1⁡ttan⁡α20tanα⁡/2=1sin⁡α2cos⁡α2⋅π4=π2sin⁡α