Sum of Cosecant and Cotangent
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Theorem
- $\csc x + \cot x = \cot {\dfrac x 2}$
Proof
| \(\ds \csc x + \cot x\) | \(=\) | \(\ds \frac 1 {\sin x} + \frac {\cos x} {\sin x}\) | Definition of Cosecant and Definition of Cotangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {1 + \cos x} {\sin x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 \cos^2 {\frac x 2} } {2 \sin {\frac x 2} \cos {\frac x 2} }\) | Double Angle Formula for Sine and Double Angle Formula for Cosine: Corollary $1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\cos {\frac x 2} } {\sin {\frac x 2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cot {\frac x 2}\) | Definition of Cotangent |
$\blacksquare$