Cosecant Minus Sine

Theorem

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


Proof

\(\ds \csc x - \sin x\) \(=\) \(\ds \frac 1 {\sin x} - \sin x\) Definition of Cosecant
\(\ds \) \(=\) \(\ds \frac {1 - \sin^2 x} {\sin x}\)
\(\ds \) \(=\) \(\ds \frac {\cos^2 x} {\sin x}\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds \cos x \ \cot x\) Definition of Cotangent

$\blacksquare$

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