Prove that:

[CBSE 2011]

To prove:

Let, y =

let, x = cos 2θ

⇒ θ = 1/2 cos^-1 x …(1)

we know that-

1 + cos 2θ = 2cos²θ

And 1 – cos 2θ = 2sin²θ

∴ y =

⇒ y =

⇒ y =

⇒ y =

⇒ y =

⇒ y =

We know that: tan(x - y) =

∴ y =

∴ y = {from 1}

From basic ITF formula we know that –

tan^-1(tan x) = x if x ∈ (-π/2 , π/2)

Given,

(-1/√2) ≤ x ≤ 1

∴ 0 ≤ cos^-1x ≤ 3π/4

⇒ -3π/8 ≤ -(1/2)cos^‑1x ≤ 0

⇒ -π/8 < π/4 – (1/2)cos^-1x ≤ π/4

Thus,

∴ y = ,