Question

1+√3tan^2(x)=(1+√3)tanx then find x​

Answer 1

Answer

The value of x is either 45°(π/4) or 30°(π/6)

Solution

Given,

$\sf1 + \sqrt{3} \tan {}^{2} x = (1 + \sqrt{3} ) \tan x \\ \\ \sf \implies \sqrt{3} \tan {}^{ 2} x - (1 + \sqrt{3} ) \tan x + 1 = 0 \\ \\ \sf \implies \sqrt{3} \tan {}^{2}x - \tan x - \sqrt{3} \tan x + 1 = 0 \\ \\ \sf \implies \sqrt{3} \tan {}^{2}x - \sqrt{3} \tan x - \tan x + 1 = 0 \\ \\ \sf\implies \sqrt{3} \tan x( \tan x - 1) - 1( \tan x - 1) = 0 \\ \\ \sf \implies( \sqrt{3} \tan x - 1)( \tan x - 1) = 0$

Now we have :

$\implies \sf\sqrt{3} \tan x - 1 = 0 \\ \\ \sf\implies \tan x = \frac{1}{ \sqrt{3} } \\ \\ \implies \sf \tan x = \tan30 \degree \\ \\ \sf\implies x = 30\degree \\ \\ \implies \sf x = \dfrac{ \pi}{6}$

and again ,

$\sf \implies \tan x - 1 = 0 \\\\ \sf\implies tan x = 1 \\\\ \sf \implies \tan x = \tan 45\degree \\\\ \sf\implies x = 45\degree \\\\ \implies \sf x = \dfrac{\pi}{4}$