Cosecant Minus Sine
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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 |
$\blacksquare$