cot−1⁡(cos⁡α)−tan−1⁡(cos⁡α)=x, then sin x =

cot−1⁡(cos⁡α)−tan−1⁡(cos⁡α)=x⇒tan−1⁡1cos⁡α−tan−1⁡(cos⁡α)=x⇒tan−1⁡1cos⁡α−cos⁡α1+1cos⁡αcos⁡α=x ⇒tan−1⁡1−cos⁡α2cos⁡α=x⇒tan⁡x=1−cos⁡α2cos⁡α or cot⁡x=2cos⁡α1−cos⁡α or cosec⁡x=1+cos⁡α1−cos⁡α∴sin⁡x=1−cos⁡α1+cos⁡α=1−1−2sin2⁡α/21+2cos2⁡α/2−1 or sin⁡x=tan2⁡α/2