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
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Exercise $\text {XXXI}$: $1.$