If 0 < x < 1/√2 and (sin ⁻¹x)/α = (cos ⁻¹x)/β, then the value of sin ( (2π α)/(α + β) ) is :

Solution

Maths sir · 4 steps

Use sin⁻¹ x + cos⁻¹ x = π/2 to show the angle is just 4 sin⁻¹ x.

1. Name the common ratio. Let (sin⁻¹ x)/α = (cos⁻¹ x)/β = k. Then α = (sin⁻¹ x)/k and β = (cos⁻¹ x)/k.
2. Add alpha and beta. α + β = (sin⁻¹ x + cos⁻¹ x)/k = π/2k. So the angle becomes 2πα/(α+β) = 2π × (sin⁻¹ x)/k × 2k/π = 4 sin⁻¹ x.
3. Expand sin 4θ. Put θ = sin⁻¹ x, so sin θ = x and cos θ = √(1-x²). Write sin 4θ = 2 sin 2θ cos 2θ with sin 2θ = 2 sin θ cos θ and cos 2θ = 1 - 2 sin² θ.

   sin 4θ = 2(2x√(1-x²))(1-2x²) = 4x√(1-x²)(1-2x²)

4. So. sin(2πα/(α+β)) = 4x√(1-x²)(1-2x²), which is option (b).

The idea to keep. When sin⁻¹ x and cos⁻¹ x appear together, their sum π/2 is almost always the key. Use it to get rid of the extra letters, then expand with double-angle formulas.

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