First slide

Trigonometric equations

Question

Number of solutions of the equation $4 (\cos^{2} 2 x + \cos 2 x + 1) + \tan x (\tan x - 2 \sqrt{3}) = 0$ in $[0, 2 π]$ is

Moderate

A

0
B

1
C

2
D

3

Solution

$\begin{array}{l} (4 \cos^{2} 2 x + 4 \cos 2 x + 1) + (\tan^{2} x - 2 \sqrt{3} \tan x + 3) = 0 \\ \Rightarrow (2 \cos 2 x + 1)^{2} + (\tan x - \sqrt{3})^{2} = 0 \\ \Rightarrow \cos 2 x = - \frac{1}{2} and \tan x = \sqrt{3} \\ \Rightarrow 2 \cos^{2} x - 1 = - \frac{1}{2} and \tan x = \sqrt{3} \\ \Rightarrow \cos x = \pm \frac{1}{2} and \tan x = \sqrt{3} \\ \Rightarrow x = \frac{π}{3}, \frac{4 π}{3} \end{array}$
$∴$ Number of solutions of equation = 2

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