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