Real sine and cosine function/Derivative/Fact

The

sine function

$ℝ⟶ℝ,x⟼\sin x,$

is differentiable, with

$\sin {}^{\prime }(x)=\cos x,$

and the cosine function

$ℝ⟶ℝ,x⟼\cos x,$

is differentiable, with

$\cos {}^{\prime }(x)=-\sin x.$

Write a proof