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Question
Prove that `3sin^(-1)x = sin^(-1) (3x - 4x^3)`, `x in [-1/2, 1/2]`
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Solution
To prove `3sin^(-1)x = sin^(-1) (3x - 4x^3)`, `x in [-1/2, 1/2]`
R.H.S : `sin^(-1) (3x - 4x^3)`
Let `x = sin theta`
`=> theta = sin^(-1)x `
Putting this value of x in RHS, we get
`= sin^(-1) (3sin theta - 4sin^3 theta)`
`= sin^(-1) (sin 3theta)` `(∵ sin 3theta = 3sintheta - 4sn^3 theta)`
`= 3theta`
`= 3sin^(-1) x = L.H.S`
Thus, LHS = RHS
Hence Proved
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