Sum of Reciprocals of One Plus and Minus Sine
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Theorem
- $\dfrac 1 {1 - \sin x} + \dfrac 1 {1 + \sin x} = 2 \sec^2 x$
Proof
\(\ds \frac 1 {1 - \sin x} + \frac 1 {1 + \sin x}\) | \(=\) | \(\ds \frac {1 + \sin x + 1 - \sin x} {1 - \sin^2 x}\) | Difference of Two Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {\cos^2 x}\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \sec^2 x\) | Definition of Secant Function |
$\blacksquare$