Reciprocal of One Plus Sine
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Theorem
- $\dfrac 1 {1 + \sin x} = \dfrac 1 2 \map {\sec^2} {\dfrac \pi 4 - \dfrac x 2}$
Proof
\(\ds 1 + \sin x\) | \(=\) | \(\ds \sin \frac \pi 2 + \sin x\) | Sine of Right Angle | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \map \sin {\frac 1 2 \paren {\frac \pi 2 + x} } \map \cos {\frac 1 2 \paren {\frac \pi 2 - x} }\) | Sine minus Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \map \sin {\frac \pi 4 + \frac x 2} \map \cos {\frac \pi 4 - \frac x 2}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \map \cos {\frac \pi 2 - \paren {\frac \pi 4 - \frac x 2} } \map \cos {\frac \pi 4 - \frac x 2}\) | Cosine of Complement equals Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \map \cos {\frac \pi 4 - \frac x 2} \map \cos {\frac \pi 4 - \frac x 2}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \map {\cos^2} {\frac \pi 4 - \frac x 2}\) | simplifying | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac 1 {1 + \sin x}\) | \(=\) | \(\ds \frac 1 2 \map {\sec^2} {\frac \pi 4 - \frac x 2}\) | Definition of Secant Function |
$\blacksquare$