Sum of zeta(2s) fractions without pi^(2s) in the numerators

Question

$$ \sum _{n=1}^{\infty } \sum _{r=1}^{\infty } (\pi r)^{-2 n}=\frac{1}{2} (1-1 \cot(1)) $$

$\frac{1}{2} (1-1 \cot(1))$ is not in OEIS, so it doesn't seem to be well known.

Q1: Would this info be of use to others?

Q2: Should I put it into OEIS?

Answer 1

To answer both questions at once, no. The identity readily follows from the facts that $$\zeta(2n)=\frac{(-1)^{n+1} B_{2n}2^{2n}\pi^{2n}}{2(2n)!} \mbox{ and }\cot(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}2^{2n}x^{2n-1}}{(2n)!}$$ so there does not seem to be any value in publishing it. In fact, I expect (but have no reference) that one of these series is derived from the other.