半角正接置換
cos  θ = cos 2  ( θ / 2 ) − sin 2  ( θ / 2 ) sin 2  ( θ / 2 ) + cos 2  ( θ / 2 ) = 1 − tan 2  ( θ / 2 ) tan 2  ( θ / 2 ) + 1 = 1 − t 2 1 + t 2 {\displaystyle \cos \theta ={\frac {\cos ^{2}(\theta /2)-\sin ^{2}(\theta /2)}{\sin ^{2}(\theta /2)+\cos ^{2}(\theta /2)}}={\frac {1-\tan ^{2}(\theta /2)}{\tan ^{2}(\theta /2)+1}}={\frac {1-t^{2}}{1+t^{2}}}}
また t を θ で微分して、
d t d θ = 1 2 cos 2  ( θ / 2 ) = 1 + tan 2  ( θ / 2 ) 2 = 1 + t 2 2 {\displaystyle {\frac {{\rm {d}}t}{{\rm {d}}\theta }}={\frac {1}{2\cos ^{2}(\theta /2)}}={\frac {1+\tan ^{2}(\theta /2)}{2}}={\frac {1+t^{2}}{2}}}
より、
d θ = 2 1 + t 2 d t {\displaystyle {\rm {d}}\theta ={\frac {2}{1+t^{2}}}\,{\rm {d}}t}