<Dd> π ⋅ cot ⁡ (π x) = lim N → ∞ ∑ n = − N N 1 x + n . (\ displaystyle \ pi \ cdot \ cot (\ pi x) = \ lim _ (N \ to \ infty) \ sum _ (n = - N) ^ (N) (\ frac (1) (x + n)).) </Dd> <P> This identity can be proven with the Herglotz trick . Combining the (--n) th with the nth term lead to absolutely convergent series: </P> <Dl> <Dd> π ⋅ cot ⁡ (π x) = 1 x + ∑ n = 1 ∞ 2 x x 2 − n 2, π sin ⁡ (π x) = 1 x + ∑ n = 1 ∞ (− 1) n 2 x x 2 − n 2 . (\ displaystyle \ pi \ cdot \ cot (\ pi x) = (\ frac (1) (x)) + \ sum _ (n = 1) ^ (\ infty) (\ frac (2x) (x ^ (2) - n ^ (2))) \, \ quad (\ frac (\ pi) (\ sin (\ pi x))) = (\ frac (1) (x)) + \ sum _ (n = 1) ^ (\ infty) (\ frac ((- 1) ^ (n) \, 2x) (x ^ (2) - n ^ (2))).) </Dd> </Dl> <Dd> π ⋅ cot ⁡ (π x) = 1 x + ∑ n = 1 ∞ 2 x x 2 − n 2, π sin ⁡ (π x) = 1 x + ∑ n = 1 ∞ (− 1) n 2 x x 2 − n 2 . (\ displaystyle \ pi \ cdot \ cot (\ pi x) = (\ frac (1) (x)) + \ sum _ (n = 1) ^ (\ infty) (\ frac (2x) (x ^ (2) - n ^ (2))) \, \ quad (\ frac (\ pi) (\ sin (\ pi x))) = (\ frac (1) (x)) + \ sum _ (n = 1) ^ (\ infty) (\ frac ((- 1) ^ (n) \, 2x) (x ^ (2) - n ^ (2))).) </Dd>

Where did sine cosine and tangent come from