Tangent times Tangent Plus Cotangent

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Theorem

$\tan x \left({\tan x + \cot x}\right) = \sec^2 x$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \tan x \left({\tan x + \cot x}\right)$	$=$	$\displaystyle \tan x \sec x \csc x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sum of Tangent and Cotangent
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {\sin x} {\cos^2 x \sin x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	By definition of tangent, secant and cosecant
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac 1 {\cos^2x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \sec^2 x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	By definition of secant

$\blacksquare$

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

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \tan x \left({\tan x + \cot x}\right)\)	\(=\)	\(\displaystyle \tan x \sec x \csc x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sum of Tangent and Cotangent
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\sin x} {\cos^2 x \sin x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	By definition of tangent, secant and cosecant
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac 1 {\cos^2x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sec^2 x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	By definition of secant

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