Reciprocal of One Plus Cosine/Proof 2
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Theorem
- $\dfrac 1 {1 + \cos x} = \dfrac 1 2 \sec^2 \dfrac x 2$
Proof
\(\ds \cos x\) | \(=\) | \(\ds 2 \cos^2 \frac x 2 - 1\) | Double Angle Formula for Cosine: Corollary $1$ | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds 1 + \cos x\) | \(=\) | \(\ds 2 \cos^2 \frac x 2\) | adding $1$ to both sides | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \frac 1 {1 + \cos x}\) | \(=\) | \(\ds \frac 1 2 \frac 1 {\cos^2 \frac x 2}\) | taking the reciprocal of both sides | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \sec^2 \frac x 2\) | Definition of Secant Function |
$\blacksquare$