Question

(sqrt3)tanx-2 sin x tan x=0 radian solutions

Answer 1

$$\sqrt{3} \tan x - 2 \sin x \cdot \tan x = 0.$$

Solution:

$$\sqrt{3} \tan x - 2 \sin x \cdot \tan x = 0,$$ $$(\sqrt{3} - 2 \sin x) \cdot \tan x = 0,$$ $$\tan x = 0, \text{ or } \sqrt{3} - 2 \sin x = 0,$$ $$x = \arctan(0) + \pi m, \quad m \in \mathbb{Z}; \quad \text{or} \quad \sin x = \frac{\sqrt{3}}{2},$$ $$x = \pi m, \quad m \in \mathbb{Z}; \quad \text{or} \quad x = (-1)^n \arcsin\left(\frac{\sqrt{3}}{2}\right) + \pi n, \quad n \in \mathbb{Z},$$ $$x = (-1)^n \frac{\pi}{3} + \pi n, \quad n \in \mathbb{Z}.$$

Answer:

$$x = \pi m, \quad m \in \mathbb{Z}$$

or

$$x = \pi \left(\frac{(-1)^n}{3} + n\right), \quad n \in \mathbb{Z}$$