Relationship between tanh, inverse Gudermannian, and sin

Theorem

The following formula holds: $$\mathrm{tanh}(\mathrm{gd}^{-1}(x))=\sin(x),$$ where $\mathrm{tanh}$ denotes the hyperbolic tangent, $\mathrm{gd}^{-1}$ denotes the inverse Gudermannian, and $\sin$ denotes sine.