# Cot X + Cot ( π 3 + X ) + Cot ( 2 π 3 + X ) = 3 Cot 3 X - Mathematics

Numerical

$\cot x + \cot\left( \frac{\pi}{3} + x \right) + \cot\left( \frac{2\pi}{3} + x \right) = 3 \cot 3x$

#### Solution

$\frac{\pi}{3} = 60° , \frac{2\pi}{3} = 120°$
$LHS = \text{ cot } x + \cot\left( 60° + x \right) + \cot\left( 120° + x \right)$
$= \text{ cot } x + \cot\left( 60° + x \right) - \cot\left[ 180° - \left( 120° + x \right) \right]$
$\left( \because - cot\theta = \cot\left( 180° - \theta \right) \right)$
$= \text{ cot } x + \cot\left( 60° + x \right) - \cot\left( 60° - x \right)$
$= \frac{1}{\text{ tan } x} + \frac{1}{\tan\left( 60° + x \right)} - \frac{1}{\tan\left( 60° - x \right)}$

$= \frac{1}{\text{ tanx } } + \frac{1 - \sqrt{3}\text{ tan } x}{\sqrt{3} + \text{ tan } x} - \frac{1 + \sqrt{3}\text{ tan } x}{\sqrt{3} - \text{ tan } x}$
$\left[ \tan\left( x + y \right) = \frac{\text{ tan } x + \text{ tan } y}{1 - \text{ tan } x\text{ tan } y} \text{ and } \tan\left( x - y \right) = \frac{\text{ tan } x - \text{ tan } y}{1 + \text{ tan } x \text{ tan } y} \right]$
$= \frac{1}{\text{ tan } x} - \frac{8\text{ tan } x}{3 - \tan^2 x}$
$= \frac{3 - \tan^2 x - 8 \tan^2 x}{3\text{ tan } x - \tan^3 x}$
$= \frac{3 - 9 \tan^2 x}{3\text{ tan } x - \tan^3 x}$
$= 3\left( \frac{1 - 3 \tan^2 x}{3\text{ tan } x - \tan^3 x} \right)$
$= 3 \times \frac{1}{\tan3x} \left( \because \tan3\theta = \frac{3tan\theta - \tan^3 \theta}{1 - 3 \tan^2 \theta} \right)$
$= 3\cot 3x$
$= RHS$
$\text{ Hence proved } .$

Concept: Values of Trigonometric Functions at Multiples and Submultiples of an Angle
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 9 Values of Trigonometric function at multiples and submultiples of an angle
Exercise 9.2 | Q 7 | Page 36