I've seen many trigonometric identities but here is one that I encountered for which I did not find a reference.

In case you wonder where this came from, I was investigating certain $q$-series in this paper on Eulerian polynomials.

Question. Is this true? Can you provide a reference or a proof? $$\sum_{k=-n}^n\frac{2k+1}{n+k+1}\binom{2n}{n-k}\sec((2k+1)\theta)=\sec((2n+1)\theta).$$