The amazing identity [1 + 2 sum_(n = 1)^∞ (cos(n θ))/(cosh(n π))]^(-2) + [1 + 2 sum_(n = 1)^∞ (cosh(n θ))/(cosh(n π))]^(-2) = (2Γ^4(3/4))/π for all θ, where Γ(z) is the gamma function. Equating coefficients of θ^0, θ^4, and θ^8 gives some amazing identities for the hyperbolic secant.

hyperbolic secant

Srinivasa Aiyangar Ramanujan