[math-fun] Identity transformation

Hi, Pi/2==sum((2*n)!*(1+x^(4*n+2))/(2^(2*n)*(n!)^2*(1+x^4)^(n+1/2)*(2*n+1)) ,n=0..inf); an identity transformation gives this result for any real number. I wonder what other process can lead to this formula Best regards.