If (sin ⁻¹x)² - (cos ⁻¹x)² = a; 0 < x < 1, a ≠ 0, then the value of 2x² - 1 is :

Solution

Maths sir · 4 steps

Factor the difference of squares; the sum sin⁻¹ x + cos⁻¹ x is always π/2.

1. Factor it. Let s = sin⁻¹ x and c = cos⁻¹ x. Then s² - c² = (s + c)(s - c), and s + c = π/2 for every x in [-1, 1].
2. Find the difference. So π/2(s - c) = a, which gives s - c = 2a/π. Subtract this from s + c = π/2 to get 2c = π/2 - 2a/π.
3. Double angle. Since x = cos c, the double-angle formula gives 2x² - 1 = 2 cos² c - 1 = cos 2c.

   2x² - 1 = cos(π/2 - 2a/π) = sin(2a/π)

4. So. 2x² - 1 = sin(2a/π), which is option (b).

The idea to keep. When you see sin⁻¹ x and cos⁻¹ x together, use their sum π/2 first. And 2x² - 1 is just cos(2 cos⁻¹ x) in disguise.

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