If tan⁻¹(x) - tan⁻¹(x+3x+4) = 34, then find the value of x.

Given tan⁻¹(x) - tan⁻¹(x+3x+4) = 34
Let tan⁻¹(x) = a and tan⁻¹(x+3x+4) = b
So we have x = tan a and x+3x+4 = tan b
The given equation will become a - b = 34
Now apply tan on both sides
We get tan(a-b) = (tan a - tan b) / (1 + tan a * tan b) = tan(34)
By substituting tan a and tan b values, we get
(x - (x+3x+4)) / (1 + x * (x+3x+4)) = tan(34)
(x - x - 3x - 4) / (1 + x² + 3x² + 4x) = tan(34)
(-3x - 4) / (1 + 4x² + 4x) = tan(34)
By solving, we get x² = 24 + 5 tan(34) / 2 + tan(34)