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Question
Solve:
sin–1(x) + sin–1(1 – x) = cos–1x.
Solution
sin–1(x) + sin–1(1 – x) = cos–1x
`\implies sin^-1(x) + sin^-1(1 - x) = π/2 - sin^-1x`
`\implies sin^-1(1 - x) = π/2 - 2sin^-1x`
`\implies (1 - x) = sin(π/2 - 2sin^-1x)`
`\implies` (1 – x) = cos (2 sin–1 x)
`\implies` (1 – x) = cos (cos–1(1 – 2x2))
`\implies` (1 – x) = 1 – 2x2
`\implies` 2x2 – x = 0
∴ x = `0, 1/2`
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