Prove that:\(\sin^{-1}(3\mathrm x -4\mathrm x^3)=3\sin^{-1}\mathrm x,

1.	Prove that:\(\sin^{-1}(3\mathrm x -4\mathrm x^3)=3\sin^{-1}\mathrm x, \|\mathrm x\|\leq\frac{1}{2}\)
Answer» To Prove: sin ^-1 (3x – 4x ³) = 3 sin ^-1 x Formula Used: sin 3A = 3 sin A – 4 sin ³A Proof: LHS = sin ^-1 (3x – 4x³) … (1) Let x = sin A … (2) Substituting (2) in (1), LHS = sin ^-1 (3 sin A – 4 sin ³ A) = sin^-1 (sin 3A) = 3A From (2), A = sin ^-1 x, 3A = 3 sin ^-1x = RHS Therefore, LHS = RHS Hence proved.

Prove that:\(\sin^{-1}(3\mathrm x -4\mathrm x^3)=3\sin^{-1}\mathrm x, |\mathrm x|\leq\frac{1}{2}\)

Answer»

To Prove:

sin ^-1 (3x – 4x ³) = 3 sin ^-1 x

Formula Used: sin 3A = 3 sin A – 4 sin ³A

Proof:

LHS = sin ^-1 (3x – 4x³) … (1)

Let x = sin A … (2)

Substituting (2) in (1),

LHS = sin ^-1 (3 sin A – 4 sin ³ A)

= sin^-1 (sin 3A)

= 3A

From (2), A = sin ^-1 x,

3A = 3 sin ^-1x

= RHS

Therefore, LHS = RHS

Hence proved.