Reciprocal of One Plus Cosine/Proof 3
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Theorem
- $\dfrac 1 {1 + \cos x} = \dfrac 1 2 \sec^2 \dfrac x 2$
Proof
| \(\ds \frac 1 {1 + \cos x}\) | \(=\) | \(\ds \frac 1 {1 + \frac {1- \tan^2 \frac x 2} {1 + \tan^2 \frac x 2} }\) | Tangent Half-Angle Substitution for Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {1 + \tan^2 \frac x 2} 2\) | multiplying through $\frac {1 + \tan^2 \frac x 2} {1 + \tan^2 \frac x 2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \sec^2 \frac x 2\) | Difference of Squares of Secant and Tangent |
$\blacksquare$