Reciprocal of One Minus Cosine plus Reciprocal of One Plus Cosine

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Theorem

$\dfrac 1 {1 - \cos x} + \dfrac 1 {1 + \cos x} = 2 \cosec^2 x$


Proof

\(\ds \dfrac 1 {1 - \cos x} + \dfrac 1 {1 + \cos x}\) \(=\) \(\ds \dfrac {\paren {1 + \cos x} + \paren {1 - \cos x} } {\paren {1 - \cos x} \paren {1 + \cos x} }\) common denominator
\(\ds \) \(=\) \(\ds \dfrac 2 {1 - \cos^2 x}\) Difference of Two Squares and simplification
\(\ds \) \(=\) \(\ds \dfrac 2 {\sin^2 x}\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds 2 \cosec^2 x\) Cosecant is Reciprocal of Sine

$\blacksquare$


Sources