Cotangent Minus Tangent

Theorem

$\cot x - \tan x = 2 \cot 2 x$

$\ds \cot x - \tan x$

$=$

$\ds \frac {\cos x} {\sin x} - \frac {\sin x} {\cos x}$

Definition of Tangent and Cotangent

$\ds $

$=$

$\ds \frac {\cos^2 x - \sin^2 x} {\sin x \cos x}$

$\ds $

$=$

$\ds 2 \frac {\cos^2 x - \sin^2 x} {2 \sin x \cos x}$

$\ds $

$=$

$\ds 2 \frac {\cos 2 x} {\sin 2 x}$

$\ds $

$=$

$\ds 2 \cot 2 x$

Definition of Cotangent

$\blacksquare$