Prove that:cos-1(1 - 2x2) = 2sin-1x

1.	Prove that:cos-1(1 - 2x2) = 2sin-1x
Answer» 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.

Answer»

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.

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