Step 1: Understand that tan^(-1)(x) is the angle whose tangent is x. Let's call this angle θ. So, θ = tan^(-1)(x).
Step 2: By the definition of tangent, we have tan(θ) = x. This means that in a right triangle, the opposite side is x and the adjacent side is 1.
Step 3: Use the Pythagorean theorem to find the hypotenuse of the triangle. The hypotenuse h can be calculated as h = √(opposite^2 + adjacent^2) = √(x^2 + 1^2) = √(x^2 + 1).
Step 4: Now, we need to find sin(θ). The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse. So, sin(θ) = opposite/hypotenuse = x/√(x^2 + 1).
Step 5: Therefore, we can conclude that sin(tan^(-1)(x)) = x/√(1 + x^2).
