Express in terms of x: sin(2arctan(x))

Question

PLEASE give clear steps! Thank You!

If the answers ends up having square roots, can you simply it where it does not

Answer 1

To visualize this, we can draw a right triangle ABC with side AB of length 1, side BC of length x and side AC of length sqrt(x^2 + 1). Angle B is a right angle and side AC is the hypotenuse. We can see that the tangent of angle A is x (tan(A) = x). Therefore arctan(x) = angle A, so sin(2arctan(x)) = sin(2*A). From the double angle formula for sin (which you can google), sin(2x) = 2 * sin(x) * cos(x), so sin(2*A) = 2 * sin(A) * cos(A). Referring back to the triangle, we can see that sin(A) = x / sqrt(x^2 + 1) and cos(A) = 1 / sqrt(x^2 + 1). Therefore sin(2*A) = 2 * sin(A) * cos(A) = 2 * (x / sqrt(x^2 + 1)) * (1 / sqrt(x^2 + 1)) = (2 * x) / (x^2 + 1).

So sin(2arctan(x)) = 2*x / (x^2 + 1).

Hope this helps! Let me know if I need to explain anything.