Tangent and Cotangent are Cofunctions

Theorem

$\text {(1)}: \quad$

$\ds \forall x \in \R, \cos x \ne 0: \, $

$\ds \tan x$

$=$

$\ds \map \cot {90 \degrees - x}$

$\text {(2)}: \quad$

$\ds \forall x \in \R, \sin x \ne 0: \, $

$\ds \cot x$

$=$

$\ds \map \tan {90 \degrees - x}$

$\tan x = \dfrac {\sin x} {\cos x}$

Hence in order for $\tan x$ to be defined it is necessary for $\cos x \ne 0$.

Then we have:

$\Box$

$\cot x = \dfrac {\cos x} {\sin x}$

Hence in order for $\cot x$ to be defined it is necessary for $\sin x \ne 0$.

Then we have:

$\Box$

Hence the result by definition of cofunction.

$\blacksquare$