Sum of Reciprocals of One Plus and Minus Sine

Theorem

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

Proof

\(\ds \frac 1 {1 - \sin x} + \frac 1 {1 + \sin x}\)

\(=\)

\(\ds \frac {1 + \sin x + 1 - \sin x} {1 - \sin^2 x}\)

Difference of Two Squares

\(\ds \)

\(=\)

\(\ds \frac 2 {\cos^2 x}\)

Sum of Squares of Sine and Cosine

\(\ds \)

\(=\)

\(\ds 2 \sec^2 x\)

Definition of Secant Function

$\blacksquare$