Final answer:
The equation tan(tan^-1(x)) = x is true for all real numbers because the inverse tangent function outputs an angle whose tangent value is the input x, and the tangent function then returns this same value of x.
Step-by-step explanation:
The equality tan(tan-1(x)) = x holds true for all real numbers for which the tangent function is defined. The function tan-1(x), also known as the inverse tangent or arctan(x), will output an angle whose tangent is x.
The original tangent function, tan(θ), then takes this angle as input and returns the ratio of the opposite side to the adjacent side of a right-angled triangle with that angle.
For the equality to hold, the domain of x must be the range of arctan(x), which is all real numbers since any real number can represent such a ratio in a triangle. Therefore, the values of x for which the equation is true is the set of all real numbers.