Sum of Squares of Secant and Cosecant

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Theorem

$\sec^2 x + \csc^2 x = \sec^2 x \csc^2 x$


Proof

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

$\blacksquare$