Cotangent Minus Tangent
Jump to navigation
Jump to search
Theorem
- $\cot x - \tan x = 2 \cot 2 x$
Proof
\(\ds \cot x - \tan x\) | \(=\) | \(\ds \frac {\cos x} {\sin x} - \frac {\sin x} {\cos x}\) | Definition of Tangent and Cotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\cos^2 x - \sin^2 x} {\sin x \cos x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \frac {\cos^2 x - \sin^2 x} {2 \sin x \cos x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \frac {\cos 2 x} {\sin 2 x}\) | Double Angle Formula for Sine and Double Angle Formula for Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \cot 2 x\) | Definition of Cotangent |
$\blacksquare$