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Question
Prove the following:
`sin^-1(1/sqrt(2)) -3sin^-1(sqrt(3)/2) = -(3π)/(4)`
Solution
Let `sin^-1(1/sqrt(2)) = α, "where" - pi/(2) ≤ α ≤ pi/(2)`
∴ sin α = `(1)/sqrt(2) = sin pi/(4)`
∴ α = `pi/(4) ...[∵ - pi/(2) ≤ pi/(4) ≤ pi/(2)]`
∴ `sin^-1(1/sqrt(2)) = pi/(4)` ...(1)
Let `sin^-1(sqrt(3)/2) = β, "where" - pi/(2) ≤ β ≤ pi/(2)`
∴ sin β = `sqrt(3)/(2) = sin pi/(3)`
∴ β = `pi/(3) ...[∵ - pi/(2) ≤ pi/(3) ≤ pi/(2)]`
∴ `sin^-1(sqrt(3)/2) = pi/(3)` ...(2)
L.H.S. = `sin^-1(1/sqrt(2)) - 3sin^-1(sqrt(3)/2)`
= `pi/(4) - 3(pi/3)` ...[By (1) and (2)]
= `pi/(4) - pi`
= `-(3pi)/(4)`
= R.H.S.
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