Cosecant Minus Sine

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Theorem

$\csc x - \sin x = \cos x \ \cot x$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \csc x - \sin x$	$=$	$\displaystyle \frac 1 {\sin x} - \sin x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of cosecant
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {1 - \sin^2 x} {\sin x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {\cos^2x} {\sin x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sum of Squares of Sine and Cosine
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \cos x \ \cot x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of cotangent

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \csc x - \sin x\)	\(=\)	\(\displaystyle \frac 1 {\sin x} - \sin x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of cosecant
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {1 - \sin^2 x} {\sin x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\cos^2x} {\sin x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sum of Squares of Sine and Cosine
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos x \ \cot x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of cotangent