tanarctanx

tanarctanx refers to the composition of the tangent function with the arctangent function. The arctangent function, denoted as arctan(x) or tan⁻¹(x), is the inverse of the tangent function. It returns the angle whose tangent is x. The domain of arctan(x) is all real numbers, and its range is (-π/2, π/2).

When the tangent function is applied to the output of the arctangent function, i.e., tan(arctan(x)), the result is simply x. This is because the tangent function and the arctangent function are inverses of each other. For any real number x, the tangent of the angle whose tangent is x is x itself. Therefore, tan(arctan(x)) = x for all real numbers x.

This identity is fundamental in calculus and trigonometry when simplifying expressions or evaluating integrals. For example, when dealing with integrals involving arctangent functions, recognizing this composition can lead to straightforward solutions. The relationship highlights the direct cancellation that occurs when an inverse function is applied to its corresponding function.