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

Prove that:

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

Formula Used: cos 2A = 2 cos² A – 1

Proof:

LHS = cos^-1 (2x² - 1) … (1)

Let x = cos A … (2)

Substituting (2) in (1),

LHS = cos^-1 (2 cos² A – 1)

= cos^-1 (cos 2A)

= 2A

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

2A = 2 cos^-1 x

= RHS

Therefore, LHS = RHS

Hence proved.