Secant Minus Cosine

Theorem

$\sec x - \cos x = \sin x \tan x$


Proof

\(\ds \sec x - \cos x\) \(=\) \(\ds \frac 1 {\cos x} - \cos x\) Secant is Reciprocal of Cosine
\(\ds \) \(=\) \(\ds \frac {1 - \cos^2 x} {\cos x}\)
\(\ds \) \(=\) \(\ds \frac {\sin^2 x} {\cos x}\) Sum of Squares of Sine and Cosine‎
\(\ds \) \(=\) \(\ds \sin x \tan x\) Tangent is Sine divided by Cosine

$\blacksquare$

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