Question

2 cos-1 x = cos-1 (2x2 -1)

Answer 1

To prove--->

2Cos⁻¹x = Cos⁻¹ ( 2x² - 1 )

Proof---> Let, Cos⁻¹x = θ

=> x = Cosθ

Now we know that,

Cos2θ = 2Cos²θ - 1

=> Cos2θ = 2 ( Cosθ )² - 1

Putting Cοsθ = x , we get,

=> Cos2θ = 2 ( x )² - 1

=> Cos2θ = 2x² - 1

Taking cos⁻¹ both sides , we get,

=> Cos⁻¹ ( Cos2θ ) = Cos⁻¹ ( 2x² - 1 )

=> 2θ = Cos⁻¹ ( 2x² - 1 )

Putting θ = Cos⁻¹x , we get,

=> 2 Cos⁻¹ x = Cos⁻¹ ( 2x² - 1 )