Tangent/Cotangent/Derivative/Fact

The

tangent function

$\mathbb {R} \setminus {\left({\frac {\pi }{2}}+\mathbb {Z} \pi \right)}\longrightarrow \mathbb {R} ,x\longmapsto \tan x,$

is differentiable, with

${}\tan \!'(x)={\frac {1}{\cos ^{2}x}}\,,$

and the cotangent function

$\mathbb {R} \setminus \mathbb {Z} \pi \longrightarrow \mathbb {R} ,x\longmapsto \cot x,$

is differentiable, with

${}\cot \!'(x)=-{\frac {1}{\sin ^{2}x}}\,.$

Proof, Write another proof