Tangent times Tangent plus Cotangent/Proof 1
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Theorem
- $\tan x \paren {\tan x + \cot x} = \sec^2 x$
Proof
\(\ds \tan x \left({\tan x + \cot x}\right)\) | \(=\) | \(\ds \tan x \sec x \csc x\) | 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 |
$\blacksquare$