Reciprocal of One Minus Secant

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Theorem

$\displaystyle \frac {\sin^2 x + 2 \cos x - 1} {\sin^2 x + 3 \cos x - 3} = \frac 1 {1 - \sec x}$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \frac {\sin^2 x + 2 \cos x - 1} {\sin^2 x + 3 \cos x - 3}$	$=$	$\displaystyle \frac {1 - \cos^2 x - 2 \cos x - 1} {1 - \cos^2 x + 3 \cos x - 3}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sum of Squares of Sine and Cosine
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {\cos^2 x - 2 \cos x} {\cos^2 x - 3 \cos x + 2}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {\cos x \left({\cos x - 2}\right)} {\left({\cos x - 1}\right) \left({\cos x - 2}\right)}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {\cos x} {\cos x - 1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac 1 {1 - \frac 1 {\cos x} }$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac 1 {1 - \sec x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of secant

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \frac {\sin^2 x + 2 \cos x - 1} {\sin^2 x + 3 \cos x - 3}\)	\(=\)	\(\displaystyle \frac {1 - \cos^2 x - 2 \cos x - 1} {1 - \cos^2 x + 3 \cos x - 3}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sum of Squares of Sine and Cosine
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\cos^2 x - 2 \cos x} {\cos^2 x - 3 \cos x + 2}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\cos x \left({\cos x - 2}\right)} {\left({\cos x - 1}\right) \left({\cos x - 2}\right)}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\cos x} {\cos x - 1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac 1 {1 - \frac 1 {\cos x} }\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac 1 {1 - \sec x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of secant