Reciprocal of One Minus Cosine/Proof 2

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

$\blacksquare$