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Question
Prove the following:
3 sin−1 x = sin−1 (3x − 4x3), `x ∈ [-1/2, 1/2]`
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Solution
Let x = sin θ
Then, sin−1 x = θ
We have
R.H.S = sin−1 (3x − 4x3) = sin−1 (3 sin θ − 4 sin3θ)
= sin−1 (sin 3θ) = sin−1 (3 sin θ − 4 sin3θ)
= 3θ = sin−1 (3 sin θ − 4 sin3θ)
= 3 sin−1 x = sin−1 (3 sin θ − 4 sin3θ)
R.H.S = L.H.S
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