Tangent/Cotangent/Derivative/Fact/Proof

Proof

Using the quotient rule, fact, and the circle equation, we get

${}{\begin{aligned}(\tan x)^{\prime }&={\left({\frac {\sin x}{\cos x}}\right)}^{\prime }\\&={\frac {(\cos x)(\cos x)-(\sin x)(-\sin x)}{\cos ^{2}x}}\\&={\frac {1}{\cos ^{2}x}}.\end{aligned}}$

The derivative of the cotangent function follows in the same way.

To fact