Secant Plus One over Secant Squared

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Theorem

$\displaystyle \frac {\sec x + 1} {\sec^2 x} = \frac {\sin^2 x} {\sec x - 1}$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \frac {\sec x + 1} {\sec^2 x}$	$=$	$\displaystyle \cos^2 x \left({\frac 1 {\cos x} + 1}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of secant
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \cos x + \cos^2x$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \cos x \left({1 + \cos x}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \cos x \frac{\left({1 + \cos x}\right) \left({1 - \cos x}\right)} {1 - \cos x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac{1 - \cos^2 x} {\frac{1 - \cos x} {\cos x} }$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Difference of Two Squares
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac{\sin^2 x} {\frac 1 {\cos x} - 1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sum of Squares of Sine and Cosine
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac{\sin^2 x} {\sec x - 1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of secant

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \frac {\sec x + 1} {\sec^2 x}\)	\(=\)	\(\displaystyle \cos^2 x \left({\frac 1 {\cos x} + 1}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of secant
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos x + \cos^2x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos x \left({1 + \cos x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos x \frac{\left({1 + \cos x}\right) \left({1 - \cos x}\right)} {1 - \cos x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac{1 - \cos^2 x} {\frac{1 - \cos x} {\cos x} }\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Difference of Two Squares
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac{\sin^2 x} {\frac 1 {\cos x} - 1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sum of Squares of Sine and Cosine
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac{\sin^2 x} {\sec x - 1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of secant