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 |
$\blacksquare$