Often times, we consult resources, like Abramowitz and Stegun's Handbook of Mathematical Functions https://www.math.ubc.ca/~cbm/aands/, NIST's database on special functions https://www.nist.gov/programs-projects/special-functions, or Mathematica to find identities which aid us with some kind of computation.

However, what if we want to know if we have found a new identity and want to add to the library? As a simple example, I found

$\lim_{N\to\infty}\sum_{j=0}^N {2j \choose j} \left(\frac{\cos(x)}{2}\right)^{2j}$

converges pointwise to $|\mathrm{csc}(x)|.$ I don't see a representation of this type in the resources provided above.

My question is: How do we add to libraries of special function identities, and are there journals which, even today, still consider mathematical effort toward discovering identities of classical functions?