Question

Can anyone provide a hint for solving the equation 2arctan (2√x/1+x) =x

Answer 1

To solve the equation 2arctan (2√x/1+x) =x, you need to isolate x. Follow these steps: multiply both sides by 1 + x, take the tangent of both sides, use the double angle identity for tangent, square both sides, expand the squared terms, and finally set up and solve an inverse sine equation.

Step-by-step explanation:

To solve the equation 2arctan (2√x/1+x) =x, we need to isolate x. Here is a step-by-step explanation:

1. Multiply both sides of the equation by 1 + x: 2arctan (2√x) = x(1 + x)
2. Take the tangent of both sides of the equation to eliminate the arctan: tan (2arctan (2√x)) = tan (x(1 + x))
3. Use the double angle identity for tangent to simplify: 2√x = tan (x + x²)
4. Square both sides of the equation to eliminate the square root: (2√x)² = (tan (x + x²))²
5. Expand the squared terms: 4x = tan² (x + x²)
6. Use the identity tan² θ = 1 - cos² θ to rewrite the equation: 4x = 1 - cos² (x + x²)
7. Use the identity cos² θ = 1 - sin² θ to rewrite the equation: 4x = sin² (x + x²)
8. Take the square root of both sides: 2√x = sin (x + x²)
9. Set up an inverse sine equation: x + x² = sin¯¹ (2√x)
10. Solve for x by using a numerical method or approximations.