Sum of Tangent and Cotangent

From ProofWiki

Jump to: navigation, search

Theorem

$\tan x + \cot x = \sec x \csc x$

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

$\blacksquare$

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