Proving The Identity: Tan^(-1)(Tan(X)) = X In Interval (-Π/2, Π/2)

tan−1(tan(x)) =.__________for x in__________

x; (−π/2,π/2)

The identity tan^(-1)(tan(x)) = x holds true for x in the interval (-π/2, π/2), where tan(x) is defined and is a one-to-one function.

Here’s a step-by-step explanation:

1. Recall that the tangent function is defined as tan(x) = sin(x)/cos(x), and is only defined when cos(x) is nonzero. Therefore, the domain of tan(x) is all real numbers except those of the form π/2 + πn, where n is an integer.

2. In the interval (-π/2, π/2), the tangent function is strictly increasing and one-to-one, which means that it has an inverse function tan^(-1)(x), also known as arctan(x). This function takes input values in the range (-∞, ∞) and outputs angles in the interval (-π/2, π/2).

3. Applying the identity tan^(-1)(tan(x)) = x involves using the definition of inverse functions. If y = tan^(-1)(tan(x)), then tan(y) = tan(tan^(-1)(tan(x))) = tan(x), since the tangent function cancels out with its inverse. But tan(x) is defined in the interval (-π/2, π/2), so y must also be in this interval. Therefore, we have y = x, which means that tan^(-1)(tan(x)) = x for x in (-π/2, π/2).

More Answers:
Discovering The Domain And Range Of Cos⁻¹(X) – A Comprehensive Guide.
Discover The Domain And Range Of Sin⁻¹(X) In Simple Steps
The Inverse Sine Function: Working With Degrees And Radians

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