Simplify the expression.tan(sin−1(x))

The expression tan(sin^(-1)(x)) can be simplified using the Pythagorean identity and the definition of tangent.

Step 1: Let's denote sin^(-1)(x) as θ. So, θ is an angle whose sine is x. By definition, sin(θ) = x.

Step 2: According to the Pythagorean identity, we know that sin^2(θ) + cos^2(θ) = 1. Since sin(θ) = x, we can substitute x into the identity: x^2 + cos^2(θ) = 1.

Step 3: Solve the equation for cos(θ): cos^2(θ) = 1 - x^2. Therefore, cos(θ) = sqrt(1 - x^2) or cos(θ) = -sqrt(1 - x^2). However, since θ = sin^(-1)(x) is defined to be in the interval [-π/2, π/2], cos(θ) is nonnegative. So, we only take the positive root: cos(θ) = sqrt(1 - x^2).

Step 4: Now, we can find tan(θ) using the definition of tangent: tan(θ) = sin(θ)/cos(θ). Substitute sin(θ) = x and cos(θ) = sqrt(1 - x^2) into the definition: tan(θ) = x/sqrt(1 - x^2).

So, tan(sin^(-1)(x)) = x/sqrt(1 - x^2).