Taylor series for Gudermannian

Theorem: The following formula holds: $$\dfrac{\mathrm{gd}(x)}{2} = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k \mathrm{tanh}^{2k+1}(\frac{x}{2})}{2k+1},$$ where $\mathrm{gd}$ is the Gudermannian and $\tanh$ is the hyperbolic tangent.

Proof: █