Question

sin-1x+sin-1y+sin-1x=π÷2 then show that x2+y2+2xyz​

Answer 1

Let sin–1x = α, sin–1y = β, sin–1z = γ. Then x = sinα, y = sinβ, z = sinγ => sin–1x + sin–1y + sin–1z = π/2 => sin–1x + sin–1y = (π/2) - sin–1z => α + β = (π/2) - γ => cos(α + β) = cos{(π/2) - γ} => cosα.cosβ - sinα.sinβ = sinγ => √(1-sin2α.√(1-sin2β) - sinα.sinβ = sinγ => √(1-x2)√(1-y2) - xy = z => √(1-x2)√(1-y2) = xy + z => (1-x2)√(1-y2) = (xy + z2)2 => 1 – y2 – x2 + x2y2 = x2y2 + z2 + 2xyz => x2 + y2 + z2 + 2xyz = 1