Secant Minus Cosine

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Theorem

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

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \sec x - \cos x$	$=$	$\displaystyle \frac 1 {\cos x} - {\cos x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of secant
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {1 - \cos^2 x} {\cos x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {\sin^2 x} {\cos x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sum of Squares of Sine and Cosine‎
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \sin x \tan x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of tangent

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \sec x - \cos x\)	\(=\)	\(\displaystyle \frac 1 {\cos x} - {\cos x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of secant
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {1 - \cos^2 x} {\cos x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\sin^2 x} {\cos x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sum of Squares of Sine and Cosine‎
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sin x \tan x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of tangent