Derivative of the Real Cosine Function

Theorem: Derivative of the Real Cosine Function

The real cosine function is differentiable everywhere and its derivative is the negative real sine function.

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

PROOF

$$(\cos x{)}^{\prime }=\sin {\left(\frac{\pi }{2}-x\right)}^{\prime }={\left(\frac{\pi }{2}-x\right)}^{\prime }\cos \left(\frac{\pi }{2}-x\right)=-\cos \left(\frac{\pi }{2}-x\right)=-\sin x$$