Relationship between cosine, Gudermannian, and sech

Theorem: The following formula holds: $$\cos(\mathrm{gd})(x)=\mathrm{sech}(x),$$ where $\cos$ denotes the cosine, $\mathrm{gd}$ denotes the Gudermannian, and $\mathrm{sech}$ denotes the hyperbolic secant.

Proof: █