Simplify tan(arctan(x))

Problem

tan(arctan(x))

Solution

1. Identify the relationship between the functions. The expression involves a trigonometric function, tan(θ) and its inverse function, arctan(x)

2. Apply the property of inverse functions. By definition, for any value x in the domain of the inverse function, ƒ*(ƒ(x)(−1))=x

3. Determine the domain. The domain of arctan(x) is all real numbers, (−∞,∞) Therefore, the composition tan(arctan(x)) is defined for all real x

4. Simplify the expression. Since tan() and arctan() are inverse operations, they undo each other.

Final Answer

tan(arctan(x))=x

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