Recently I have obtained the following:

$$ \prod_{k=1}^{\infty} \left(1 + \frac 1{4\cdot k^2\cdot(4\cdot k-3)}\right)\,\ =\,\ \frac 4\pi $$ or, equivalently, $$ \prod_{k=1}^{\infty} \left(1 - \frac 1{(2\cdot k-1)^2\cdot(4\cdot k+1))}\right)\,\ =\,\ \frac \pi 4 $$

where I have used the Wallis product. (These equations sound so classical that they must be well-known; I'd appreciate a reference.)

**QUESTION** Do you know or can you provide simple proofs of the above equation which do not use the Wallis product?

**REMARK 1** Perhaps different proofs of Wallis theorem may lead to different proofs of the above formulas (e.g. via Fourier analysis?).

**REMARK 2** In general, products $\ \prod_{k=1}^n (1+a_k)\ $ (where $\ a_k\ $ are small) can be treated by the Euler's method by studying $\ \sum_{k=1}^n\log(a_k),\ $ etc.