Reciprocal of One Minus Cosine/Proof 1

Theorem

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

Proof

\(\ds 1 - \cos x\)

\(=\)

\(\ds \cos 0 - \cos x\)

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

$\blacksquare$