Reciprocal of One Plus Cosine/Proof 2

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

$\blacksquare$