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Question
Prove the following:
`3sin^(-1) x = sin^(-1)(3x - 4x^3), x in [-1/2, 1/2]`
Solution
To prove : `3sin^(-1) x = sin^(-1)(3x - 4x^3), x in [-1/2, 1/2]`
Let x = sinθ. Then, `sin^(-1) x = 0`
We have
R.H.S = `sin^(-1) (3x - 4x^3) = sin^(-1) (3sin theta - 4 sin^3 theta)`
`= sin^(-1) (sin 3theta) = sin^(-1) (3sin theta - 4 sin^3theta)`
= `3 theta = sin^(-1) (3sin theta - 4 sin^3theta)`
`= 3 sin^(-1) x = sin^(-1) (3sin theta - 4 sin^3theta)`
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