Question

Solve:

sin^–1 (x) + sin^–1 (1 – x) = cos^–1 x

Answer 1

sin^–1 (x) + sin^–1 (1 – x) = cos^–1 x

⇒ `sin^-1(x) + sin^-1(1 - x) = π/2 - sin^-1x`

⇒ `sin^-1(1 - x) = π/2 - 2sin^-1x`

⇒ `(1 - x) = sin(π/2 - 2sin^-1x)`

⇒ (1 – x) = cos (2 sin^–1 x)

⇒ (1 – x) = cos (cos^–1 (1 – 2x²))

⇒ (1 – x) = 1 – 2x²

⇒ 2x² – x = 0

∴ x = `0, 1/2`