Reciprocal of One Plus Cosine

From ProofWiki
Jump to navigation Jump to search

Theorem

$\dfrac 1 {1 + \cos x} = \dfrac 1 2 \sec^2 \dfrac x 2$


Proof 1

\(\ds 1 + \cos x\) \(=\) \(\ds \cos 0 + \cos x\) Cosine of Zero is One
\(\ds \) \(=\) \(\ds 2 \map \cos {\dfrac {0 + x} 2} \map \cos {\dfrac {0 - x} 2}\) Cosine plus Cosine
\(\ds \) \(=\) \(\ds 2 \map \cos {\dfrac x 2} \map \cos {\dfrac {-x} 2}\) simplifying
\(\ds \) \(=\) \(\ds 2 \map \cos {\dfrac x 2} \map \cos {\dfrac x 2}\) Cosine Function is Even
\(\ds \) \(=\) \(\ds 2 \map {\cos^2} {\frac x 2}\) simplifying
\(\ds \leadsto \ \ \) \(\ds \frac 1 {1 + \cos x}\) \(=\) \(\ds \frac 1 2 \map {\sec^2} {\frac x 2}\) Definition of Secant Function

$\blacksquare$


Proof 2

\(\ds \cos x\) \(=\) \(\ds 2 \cos^2 \frac x 2 - 1\) 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

$\blacksquare$


Proof 3

\(\ds \frac 1 {1 + \cos x}\) \(=\) \(\ds \frac 1 {1 + \frac {1- \tan^2 \frac x 2} {1 + \tan^2 \frac x 2} }\) 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

$\blacksquare$


Also see