Question

Find the values of x which satisfy the equation sin^–1x + sin^–1(1 – x) = cos^–1x.

Answer 1

From the given equation

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

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

⇒ `xsqrt(1 - (1 - x)^2) + (1 - x) sqrt(1 - x^2) = sqrt(1 - x^2)`

⇒ `xsqrt(2x - x^2) + sqrt(1 - x^2) (1 - x - 1)` = 0

⇒ `x(sqrt(2x - x^2) - sqrt(1 - x^2))` = 0

⇒ x = 0 or `2x - x^2 = 1 - x^2`

⇒ x = 0 or x =`1/2`.