Tangent times Tangent plus Cotangent/Proof 2

Theorem

$\tan x \paren {\tan x + \cot x} = \sec^2 x$

$\ds \tan x \paren {\tan x + \cot x}$

$=$

$\ds \frac {\sin x} {\cos x} \paren {\frac {\sin x} {\cos x} + \frac {\cos x} {\sin x} }$

Definition of Tangent Function and Definition of Cotangent

$\ds $

$=$

$\ds \frac {\sin x} {\cos x} \paren {\frac {\sin^2 x + \cos^2 x} {\cos x \sin x} }$

$\ds $

$=$

$\ds \frac {\sin x} {\cos x} \paren {\frac 1 {\cos x \sin x} }$

$\ds $

$=$

$\ds \frac 1 {\cos^2 x}$

$\ds $

$=$

$\ds \sec^2 x$

$\blacksquare$