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  4. The number of solutions of the equation √13-18 tan x=6 tan x-3, where -2 π < x < 2 π is

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Q. The number of solutions of the equation $\sqrt{13-18 \tan x}=6 \tan x-3$, where $-2 \pi < x < 2 \pi$ is

Trigonometric Functions

Solution:

$\sqrt{13-18 \tan x}=6 \tan x-3-2 \pi < x < 2 \pi$ (given)
$\Rightarrow 13-18 \tan x=36 \tan ^{2} x+9-36 \tan x$
$\Rightarrow 36 \tan ^{2} x-18 \tan x-4=0$
$\Rightarrow 36 \tan ^{2} x-24 \tan x+6 \tan x-4=0$
$\Rightarrow 6 \tan x(6 \tan x+1)-4(6 \tan x+1)=0$
$ \tan x=\frac{2}{3}$
or $\tan x=-1 / 6$ (not possible)
Let $\tan \alpha=2 / 3$
then $\tan x=\tan \alpha$
or $x=\tan (2 / 3), \pi+\tan ^{-1}(2 / 3), \tan ^{-1}(2 / 3)-\pi, \tan ^{-1}(2 / 3)-2 \pi$