If tan⁡(θ/2)=cosec⁡θ−sin⁡θ, then cos2⁡(θ/2) is equal to

sin⁡(θ/2)cos⁡(θ/2)=1−sin2⁡θsin⁡θ⇒2sin2⁡(θ/2)=cos2⁡θ⇒cos2⁡θ+cos⁡θ−1=0⇒cos⁡θ=−1±1+42=−1±52But cos⁡θ≠−1−52  so  cos⁡θ=5−12⇒2cos2⁡(θ/2)=5−12+1=5+12⇒cos2⁡(θ/2)=5+14=cos⁡368.