# How do you verify cotx-tanx=2cot2x?

Jul 24, 2015

Use the double angle formula for $\tan \left(2 x\right)$ and the fact that $\cot \left(\theta\right) = \frac{1}{\tan} \left(\theta\right)$

#### Explanation:

Double angle formula for $\tan$
$\textcolor{w h i t e}{\text{XXXX}}$$\tan \left(2 x\right) = \frac{2 \tan \left(x\right)}{1 - {\tan}^{2} \left(x\right)}$

$2 \cot \left(2 x\right) = 2 \cdot \frac{1}{\tan} \left(2 x\right)$

$\textcolor{w h i t e}{\text{XXXX}}$$= 2 \cdot \frac{1 - {\tan}^{2} \left(x\right)}{2 \tan \left(x\right)}$

$\textcolor{w h i t e}{\text{XXXX}}$$= \frac{1}{\tan} \left(x\right) - {\tan}^{2} \frac{x}{\tan} \left(x\right)$

$\textcolor{w h i t e}{\text{XXXX}}$$= \cot \left(x\right) - \tan \left(x\right)$

Apr 6, 2017

Using the relationship between tan/cot and sin-cos, plus the double angle formulae for sin and cos. (as requested)

#### Explanation:

Remember:

$\textcolor{red}{\text{Basic definitions:}}$
color(white)("XXX")color(red)(tan(theta)=sin(theta)/cos(theta)color(white)("XXX")cot(theta)=cos(theta)/sin(theta))

$\textcolor{b l u e}{\text{Double angle formulae for sin and cos}}$
color(white)("XX"color(blue)(sin(2theta)=2 * sin(theta) * cos(theta)color(white)("XX")cos(2theta)=cos^2(theta)-sin^2(theta))

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Required to Prove:

color(green)(cot(x)-tan(x)=2cot(2x)

Proof:

$R . S .$
$\textcolor{w h i t e}{\text{XXX}} = \textcolor{g r e e n}{2 \cot \left(2 x\right)}$

$\textcolor{w h i t e}{\text{XXX}} = 2 \cdot \cos \frac{2 x}{\sin} \left(2 x\right)$

$\textcolor{w h i t e}{\text{XXX}} = \frac{\cancel{2} \cdot \left({\cos}^{2} \left(x\right) - {\sin}^{2} \left(x\right)\right)}{\cancel{2} \cdot \sin \left(x\right) \cdot \cos \left(x\right)}$

$\textcolor{w h i t e}{\text{XXX}} = {\cos}^{2} \frac{x}{\sin \left(x\right) \cdot \cos \left(x\right)} - {\sin}^{2} \frac{x}{\sin \left(x\right) \cdot \cos \left(x\right)}$

$\textcolor{w h i t e}{\text{XXX}} = \cos \frac{x}{\sin} \left(x\right) - \sin \frac{x}{\cos} \left(x\right)$

$\textcolor{w h i t e}{\text{XXX}} = \textcolor{g r e e n}{\cot \left(x\right) - \tan \left(x\right)}$

$\textcolor{w h i t e}{\text{XXX}} = L . S .$