Reciprocal of One Plus Cosine/Proof 1

Theorem

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

$\ds 1 + \cos x$

$=$

$\ds \cos 0 + \cos x$

$\ds $

$=$

$\ds 2 \map \cos {\dfrac {0 + x} 2} \map \cos {\dfrac {0 - x} 2}$

$\ds $

$=$

$\ds 2 \map \cos {\dfrac x 2} \map \cos {\dfrac {-x} 2}$

simplifying

$\ds $

$=$

$\ds 2 \map \cos {\dfrac x 2} \map \cos {\dfrac x 2}$

$\ds $

$=$

$\ds 2 \map {\cos^2} {\frac x 2}$

simplifying

$\ds \leadsto \ \ $

$\ds \frac 1 {1 + \cos x}$

$=$

$\ds \frac 1 2 \map {\sec^2} {\frac x 2}$

$\blacksquare$