Real sine and cosine function/Derivative/Fact

The

sine function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \sin x,$

is differentiable, with

${}\sin \!'(x)=\cos x\,,$

and the cosine function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \cos x,$

is differentiable, with

${}\cos \!'(x)=-\sin x\,.$

Write a proof