To Prove: cos -1 (1 – 2x 2 ) = 2 sin -1 x Formula Used: cos 2A = 1 – 2 sin 2 A Proof: LHS = cos -1 (1 – 2x 2 ) … (1) Let x = sin A … (2) Substituting (2) in (1), LHS = cos -1 (1 – 2 sin 2 A) = cos -1 (cos 2A) = 2A From (2), A = sin -1 x, 2A = 2 sin -1 x = RHS Therefore, LHS = RHS Hence proved.

Prove that:

To Prove: cos^-1 (1 – 2x²) = 2 sin^-1 x

Formula Used: cos 2A = 1 – 2 sin² A

Proof:

LHS = cos^-1 (1 – 2x²) … (1)

Let x = sin A … (2)

Substituting (2) in (1),

LHS = cos^-1 (1 – 2 sin² A)

= cos^-1 (cos 2A)

= 2A

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

2A = 2 sin^-1 x

= RHS

Therefore, LHS = RHS

Hence proved.