Prove that:

OR

If find the value of x.

Put x = cos2θ

We know that cos2θ = 1 – 2sin²θ and cos2θ = 2cos²θ – 1

Hence 1 + cos2θ = 2cos²θ and 1 – cos2θ = 2sin²θ

⇒ 1 + x = 1 + cos2θ = 2cos²θ

⇒ 1 – x = 1 – cos2θ = 2sin²θ

Consider LHS

Cancel out √2

We know that

Using

As x = cos2θ ⇒ cos^-1x = 2θ ⇒ θ = 1/2(cos^-1x)

⇒ LHS = RHS

Hence proved

OR

We know that

Here and

As

⇒ x² + 4x – 2x – 8 + x² – 4x + 2x – 8 = x² – 16 – (x² – 4)

⇒ 2x² – 16 = x² – 16 – x² + 4

⇒ 2x² = 4

⇒ x² = 2

⇒ x = ±√2

Hence x is ±√2.