Reciprocal of One Minus Cosine

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Theorem

$\dfrac 1 {1 - \cos x} = \dfrac 1 2 \map {\csc^2} {\dfrac x 2}$


Proof 1

\(\ds 1 - \cos x\) \(=\) \(\ds \cos 0 - \cos x\) Cosine of Zero is One
\(\ds \) \(=\) \(\ds -2 \map \sin {\dfrac {0 + x} 2} \map \sin {\dfrac {0 - x} 2}\) Cosine minus Cosine
\(\ds \) \(=\) \(\ds -2 \map \sin {\dfrac x 2} \map \sin {\dfrac {-x} 2}\) simplifying
\(\ds \) \(=\) \(\ds 2 \map \sin {\dfrac x 2} \map \sin {\dfrac x 2}\) Sine Function is Odd
\(\ds \) \(=\) \(\ds 2 \map {\sin^2} {\frac x 2}\) simplifying
\(\ds \leadsto \ \ \) \(\ds \frac 1 {1 - \cos x}\) \(=\) \(\ds \frac 1 2 \map {\csc^2} {\frac x 2}\) Definition of Cosecant

$\blacksquare$


Proof 2

\(\ds \cos x\) \(=\) \(\ds 1 - 2 \sin^2 \frac x 2\) 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

$\blacksquare$


Proof 3

\(\ds \frac 1 {1 - \cos x}\) \(=\) \(\ds \frac 1 {1 - \frac {1 - \map {\tan^2} {\frac x 2} } {1 + \map {\tan^2} {\frac x 2} } }\) 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

$\blacksquare$


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