Reciprocal of One Minus Cosine/Proof 3
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Theorem
- $\dfrac 1 {1 - \cos x} = \dfrac 1 2 \map {\csc^2} {\dfrac x 2}$
Proof
\(\ds \frac 1 {1 - \cos x}\) | \(=\) | \(\ds \frac 1 {1 - \frac {1 - \map {\tan^2} {\frac x 2} } {1 + \map {\tan^2} {\frac x 2} } }\) | Tangent Half-Angle Substitution for Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {1 + \map {\tan^2} {\frac x 2} } {1 + \map {\tan^2} {\frac x 2} - 1 + \map {\tan^2} {\frac x 2} }\) | multiplying top and bottom by $1 + \map {\tan^2} {\dfrac x 2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map {\sec^2} {\frac x 2} } {2 \map {\tan^2} {\frac x 2} }\) | Difference of Squares of Secant and Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \cdot \frac 1 {\map {\cos^2} {\frac x 2} } \cdot \frac {\map {\cos^2} {\frac x 2} } {\map {\sin^2} {\frac x 2} }\) | Definition of Secant Function, Definition of Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 \map {\sin^2} {\frac x 2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \map {\csc^2} {\frac x 2}\) | Definition of Cosecant |
$\blacksquare$