Question

Prove that `3sin^(-1)x = sin^(-1) (3x - 4x^3)`, `x in [-1/2, 1/2]`

Answer 1

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