Question

The smallest value of x satisfying the equation √3(cotx+tanx)=4is

Answer 1

x=π/6 or π/3 satisfies the equation

Answer 2

$Given \:equation : \sqrt{3}(cot x + tan x ) = 4$

$\implies \sqrt{3} \Big( \frac{1}{tan x} + tan x \Big) = 4$

$\implies \sqrt{3} \Big( \frac{1+tan^{2} x}{tan x} \Big) = 4$

$\implies \sqrt{3} \Big( \frac{sec^{2} x}{tan x} \Big) = 4$

$\implies \sqrt{3} sec^{2} x = 4 tan x$

$\implies \sqrt{3} \frac{1}{cos^{2} x } = 4\times \frac{sin x}{cos x}$

$\implies \sqrt{3} =2 \times 2 sin x cos x$

$\implies \frac{\sqrt{3}}{2} = sin 2x$

$\boxed { \pink {Since, 2sin x cos x = sin 2x }}$

$\implies sin \frac{\pi}{3} = 2x$

$\implies 2x = n\pi + (-1)^{n} \frac{\pi}{3} , n \in Z$

$\implies x = \frac{n\pi}{2} + (-1)^{n} \frac{\pi}{6} , n \in Z$

/* To get the smallest value of x , we put n = 0 in the above equation , we get */

$x = \frac{\pi}{6}$

Therefore.,

$\red { The \: smallest \: value \: x } \green {= \frac{\pi}{6}}$

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