Considering only the principal values of inverse functions, the set A = {x ≥ 0: tan⁻¹(2x) + tan⁻¹(3x) = π/4} Contains more than two elements, Contains two elements, is a singleton, is an empty set

Let the given equation be
tan⁻¹(2x) + tan⁻¹(3x) = π/4
Using the formula tan⁻¹(a) + tan⁻¹(b) = tan⁻¹((a+b)/(1-ab)), we get
tan⁻¹((2x + 3x)/(1 - (2x)(3x))) = π/4
tan⁻¹(5x/(1 - 6x²)) = π/4
Taking the tangent on both sides,
5x/(1 - 6x²) = tan(π/4)
5x/(1 - 6x²) = 1
5x = 1 - 6x²
6x² + 5x - 1 = 0
This is a quadratic equation in x. We can solve it using the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
where a = 6, b = 5, and c = -1.
x = (-5 ± √(5² - 4 * 6 * (-1))) / (2 * 6)
x = (-5 ± √(25 + 24)) / 12
x = (-5 ± √49) / 12
x = (-5 ± 7) / 12
x₁ = (2)/12 = 1/6
x₂ = (-12)/12 = -1
Since x ≥ 0, we have x = 1/6.
Therefore, the set A contains only one element, which is x = 1/6. Thus, the set A is a singleton.