Reciprocal of One Minus Cosine/Proof 2
Jump to navigation
Jump to search
Theorem
- $\dfrac 1 {1 - \cos x} = \dfrac 1 2 \map {\csc^2} {\dfrac x 2}$
Proof
\(\ds \cos x\) | \(=\) | \(\ds 1 - 2 \sin^2 \frac x 2\) | Double Angle Formula for Cosine: Corollary $2$ | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds 1 - \cos x\) | \(=\) | \(\ds 2 \sin^2 \frac x 2\) | rearranging | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \frac 1 {1 - \cos x}\) | \(=\) | \(\ds \frac 1 2 \frac 1 {\sin^2 \frac x 2}\) | taking the reciprocal of both sides | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \csc^2 \frac x 2\) | Definition of Cosecant |
$\blacksquare$