Cotangent Minus Tangent

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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

$\blacksquare$