Interesting relation derived from the identity $\sin^2 x + \cos^2 x \equiv 1$

Note that $(2c)!\sigma(c)=\sum_{n=0}^{c-1}\binom{2c}{2n+1}$ while $(2c)!\gamma(c)=\sum_{m=0}^c\binom{2c}{2m}$, so it reduces to the famous result that even-sized subsets of a given nonempty set of finite size are exactly as numerous as the odd-sized subsets.

Incidentally, this mysterious-looking connection of a trigonometry problem to a combinatorics problem makes much more sense if we bring in complex numbers. The Pythagorean identity is then the claim $\exp(z)\exp(-z)=1$ with $z:=ix$, i.e. $\sum_{k+l=n}\frac{(-1)^k}{k!l!}=\delta_{n0}$ if we equate $z^n$ coefficients. Multiplying by $n!$ restates the problem as $\sum_{k+l=n}(-1)^k\binom{n}{k}=0$ for $n>0$, which is just the even-minus-odd calculation.