Find the Domain tan(arcsin(x))

Problem

tan(arcsin(x))

Solution

1. Identify the inner function, which is arcsin(x) The domain of arcsin(x) is restricted to the interval [−1,1]

2. Determine the constraint for the outer function, tan(θ) The tangent function is undefined when its argument θ is an odd multiple of π/2

3. Set up the inequality for the argument of the tangent function. We must ensure that arcsin(x)≠π/2 and arcsin(x)≠−π/2

4. Solve for x by applying the sine function to both sides of the exclusions. Since sin(π/2)=1 and sin(−π/2)=−1 we find that x≠1 and x≠−1

5. Combine the initial domain of the arcsine function with the exclusions found in the previous step. The interval [−1,1] excluding the endpoints −1 and 1 results in the open interval (−1,1)

Final Answer

Domain:(−1,1)

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