Relationship between coth and csch

Theorem: The following formula holds: $$\mathrm{coth} \left( \dfrac{z}{2} \right) - \mathrm{coth}(z) = \mathrm{csch}(z),$$ where $\mathrm{coth}$ denotes the hyperbolic cotangent and $\mathrm{csch}$ denotes the hyperbolic cosecant.

Proof: █