4 tan x - 12 cot x = 0
divide by 4 and multiply by tan x
tan^2 x - 3 = 0
tan x = ±√3
x = kpi ± pi/3 for integer k
To solve the equation 4 tan x - 12 cot x = 0, we can start by simplifying the equation using trigonometric identities.
First, we know that cot x is the reciprocal of tan x, so we can rewrite the equation as 4 tan x - 12 (1/tan x) = 0.
Next, we can find the common denominator of the terms. The common denominator is tan x, so we multiply the first term by tan x to get 4 tan^2 x - 12 = 0.
Now, we have a quadratic equation in terms of tan x. To solve for tan x, we can factor the equation as (2 tan x + 6)(2 tan x - 2) = 0.
Setting each factor equal to zero, we have two possible solutions:
2 tan x + 6 = 0
Solving for tan x, we subtract 6 from both sides:
2 tan x = -6
tan x = -3
tan x = -3 is one possible solution for x.
The second equation is:
2 tan x - 2 = 0
Solving for tan x, we add 2 to both sides:
2 tan x = 2
tan x = 1
tan x = 1 is the second possible solution for x.
Therefore, the solutions to the equation 4 tan x - 12 cot x = 0 are x = -3 and x = 1.