余接定理
cot  ( u + v + w ) = cot  u + cot  v + cot  w − cot  u cot  v cot  w 1 − cot  u cot  v − cot  v cot  w − cot  w cot  u . {\displaystyle \cot(u+v+w)={\frac {\cot u+\cot v+\cot w-\cot u\cot v\cot w}{1-\cot u\cot v-\cot v\cot w-\cot w\cot u}}.}
u = α 2 {\displaystyle u={\frac {\alpha }{2}}} , v = β 2 {\displaystyle v={\frac {\beta }{2}}} , w = γ 2 {\displaystyle w={\frac {\gamma }{2}}} とすると、cot(α/2 + β/2 + γ/2) = cot π/2 = 0より、
cot  ( α 2 ) cot  ( β 2 ) cot  ( γ 2 ) = cot  ( α 2 ) + cot  ( β 2 ) + cot  ( γ 2 ) .