On the logarithmic derivative of a theta function and a fundamental identity of Ramanujan (Q1260915)

We derive a fundamental identity of Ramanujan from the identity \(\bigl( {{f'} \over f}\bigr)'= {{f''} \over f}- \bigl({{f'} \over f}\bigr)^ 2\); from this persective, this identity of Ramanujan becomes an analogue of the familiar trigonometric identity \(1+\cot^ 2\theta=\cos^ 2\theta\) and many quantities associated with this Ramanujan's identity can be seen as natural extensions of their trigonometric counterparts.