For x ∈(-1,1], the number of solutions of the equation sin ⁻¹ x=2 tan ⁻¹ x is equal to __________.

Solution

Maths sir · 4 steps

Turn 2 tan⁻¹ x into a sin⁻¹, then compare the insides, and watch the interval at the end.

1. Rewrite the right side. For |x| ≤ 1, the value 2 tan⁻¹ x lies in [-π/2, π/2], so the formula 2 tan⁻¹ x = sin⁻¹ 2x/(1+x²) holds on the whole interval.
2. Compare the insides. Now sin⁻¹ x = sin⁻¹ 2x/(1+x²). Since sin⁻¹ is one-one, the two arguments must be equal.

   x = 2x/(1+x²) ⇒ x(1+x²) = 2x ⇒ x(x-1)(x+1) = 0

3. Check the interval. The candidates are x = 0, 1, -1. The interval (-1, 1] is open at -1, so x = -1 is out. At x = 1, both sides equal π/2, and at x = 0 both sides are 0.
4. So. The solutions are x = 0 and x = 1, so the number of solutions is 2.

The common slip: 3. Keeping x = -1 gives 3, but the interval (-1, 1] is open at -1, so that root is not allowed.

The idea to keep. Before using 2 tan⁻¹ x = sin⁻¹ 2x/(1+x²), check that |x| ≤ 1. After solving, always test each root against the given interval.

Answer from the NTA final answer key. Solution by PrepWiser.