Reciprocal of One Plus Cosecant

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Theorem

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

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

$\blacksquare$

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