Secant Plus One over Secant Squared

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Theorem

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


Proof

\(\ds \frac {\sec x + 1} {\sec^2 x}\) \(=\) \(\ds \cos^2 x \paren {\frac 1 {\cos x} + 1}\) Definition of Secant Function
\(\ds \) \(=\) \(\ds \cos x + \cos^2x\)
\(\ds \) \(=\) \(\ds \cos x \paren {1 + \cos x}\)
\(\ds \) \(=\) \(\ds \cos x \frac {\paren {1 + \cos x} \paren {1 - \cos x} } {1 - \cos x}\)
\(\ds \) \(=\) \(\ds \frac {1 - \cos^2 x} {\frac {1 - \cos x} {\cos x} }\) Difference of Two Squares
\(\ds \) \(=\) \(\ds \frac {\sin^2 x} {\frac 1 {\cos x} - 1}\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds \frac {\sin^2 x} {\sec x - 1}\) Definition of Secant Function

$\blacksquare$