Tangent times Tangent plus Cotangent
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Theorem
- $\tan x \paren {\tan x + \cot x} = \sec^2 x$
where $\tan$, $\cot$ and $\sec$ denote tangent, cotangent and secant respectively.
Proof 1
| \(\ds \tan x \left({\tan x + \cot x}\right)\) | \(=\) | \(\ds \tan x \sec x \csc x\) | Sum of Tangent and Cotangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin x} {\cos x} \sec x \csc x\) | Tangent is Sine divided by Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin x} {\cos^2 x} \csc x\) | Secant is Reciprocal of Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin x} {\cos^2 x \sin x}\) | Cosecant is Reciprocal of Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {\cos^2 x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sec^2 x\) | Secant is Reciprocal of Cosine |
$\blacksquare$
Proof 2
| \(\ds \tan x \paren {\tan x + \cot x}\) | \(=\) | \(\ds \frac {\sin x} {\cos x} \paren {\frac {\sin x} {\cos x} + \frac {\cos x} {\sin x} }\) | Definition of Tangent Function and Definition of Cotangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin x} {\cos x} \paren {\frac {\sin^2 x + \cos^2 x} {\cos x \sin x} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin x} {\cos x} \paren {\frac 1 {\cos x \sin x} }\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {\cos^2 x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sec^2 x\) | Definition of Secant Function |
$\blacksquare$