Sum of Squares of Secant and Cosecant

From ProofWiki

Jump to: navigation, search

Theorem

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

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \sec^2 x + \csc^2 x$	$=$	$\displaystyle \frac 1 {\cos^2 x} + \frac 1 {\sin^2 x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of secant and cosecant
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {\sin^2 x + \cos^2 x} {\cos^2 x \ \sin^2 x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac 1 {\cos^2 x \ \sin^2 x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sum of Squares of Sine and Cosine
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \sec^2 x \ \csc^2 x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of secant and cosecant

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \sec^2 x + \csc^2 x\)	\(=\)	\(\displaystyle \frac 1 {\cos^2 x} + \frac 1 {\sin^2 x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of secant and cosecant
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\sin^2 x + \cos^2 x} {\cos^2 x \ \sin^2 x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac 1 {\cos^2 x \ \sin^2 x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sum of Squares of Sine and Cosine
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sec^2 x \ \csc^2 x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of secant and cosecant