For x∈(−1,1)], the number of solutions of the equation arcsin x = 2 arctan is equal to ____.

To find the number of solutions for the equation arcsin x = 2 arctan x where x is in the interval (−1, 1], we need to analyze both sides of the equation.

Understanding the Functions

The function arcsin x is defined for x in the range (−1, 1) and outputs values in [−π/2, π/2]. On the other hand, arctan x is defined for all real numbers and outputs values in (−π/2, π/2).

Behavior of the Functions

• arcsin x: Increases from −π/2 to π/2 as x goes from −1 to 1.
• 2 arctan x: Also increases, but its range is (−π, π) as x varies over the same interval.

Finding Intersections

To determine the number of solutions, we can graph both functions or analyze their values at key points:

• At x = 0: arcsin(0) = 0 and 2 arctan(0) = 0.
• At x = 1: arcsin(1) = π/2 and 2 arctan(1) = π.
• At x = -1: arcsin(-1) = -π/2 and 2 arctan(-1) = -π.

Conclusion on Solutions

Since both functions are continuous and strictly increasing in the interval (−1, 1], they will intersect at exactly one point. Therefore, the number of solutions to the equation arcsin x = 2 arctan x is equal to 1.