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

$\blacksquare$


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