Final answer:
The equation (6tan²x-2)(tan x²-3)=0 is solved using the zero-product property, with separate solutions for the factors 6tan²x - 2 and tan x² - 3, leading to tanx = ±√(1/3) and x = ±√(tan⁻¹(3)).
Step-by-step explanation:
The equation presented by the student is (6tan²x-2)(tan x²-3)=0. To solve this, we will use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So we can set each factor equal to zero and solve them separately:
- 6tan²x - 2 = 0
- tan x² - 3 = 0
For the first factor 6tan²x - 2 = 0, we add 2 to both sides and then divide by 6:
- 6tan²x = 2
- tan²x = 2/6
- tan²x = 1/3
- tanx = ±√(1/3)
For the second factor tan x² - 3 = 0, we add 3 to both sides:
- tan x² = 3
- x² = tan⁻¹(3)
- x = ±√(tan⁻¹(3))
Thus, the solutions to the equation are the values of x for which tanx = ±√(1/3) and those for which x = ±√(tan⁻¹(3)). Remember that since this involves the tangent function, the solutions will be periodic, complicated by the fact that we have squared the variable x and the inverse tangent function tan⁻¹(3) which would yield an angle. The exact values would depend on the interval within which you want to find the solutions.