Derivative of Tangent Function

Theorem
$$D_x \left({\tan x}\right) = \sec^2 x = \frac 1 {\cos^2 x}$$, when $$\cos x \ne 0$$.

Proof

 * From the definition of the tangent function, $$\tan x = \frac {\sin x} {\cos x}$$.
 * From Derivative of Sine Function we have $$D_x \left({\sin x}\right) = \cos x$$.
 * From Derivative of Cosine Function we have $$D_x \left({\cos x}\right) = -\sin x$$.

Then:

$$ $$ $$

This is valid only when $$\cos x \ne 0$$.

The result follows from the definition of the secant function.