Reciprocal of One Minus Cosine/Proof 3

Theorem

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

$\ds \frac 1 {1 - \cos x}$

$=$

$\ds \frac 1 {1 - \frac {1 - \map {\tan^2} {\frac x 2} } {1 + \map {\tan^2} {\frac x 2} } }$

$\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 $

$=$

$\ds \frac {\map {\sec^2} {\frac x 2} } {2 \map {\tan^2} {\frac x 2} }$

$\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

$\blacksquare$