Choose the correct answer

If , then

We are given with,

…(i)

We need to find the value of

Take Left Hand Side (LHS) of equation (i),

Using the property of inverse trigonometry,

Putting and ,

Equate LHS to RHS.

Taking cosine on both sides,

Using property of inverse trigonometry,

cos(cos^{-1} A) = A

Simplifying the equation,

Squaring on both sides,

Using algebraic identity,

(A – B)^{2} = A^{2} + B^{2} – 2AB

Using trigonometric identity,

cos 2θ = cos^{2} θ – sin^{2} θ …(ii)

sin^{2} θ + cos^{2} θ = 1 ⇒ sin^{2} θ = 1 – cos^{2} θ …(iii)

Putting value of sin^{2} θ from equation (iii) in equation (ii), we get

cos 2θ = cos^{2} θ – (1 – cos^{2} θ)

Or, cos 2θ = cos^{2} θ – 1 + cos^{2} θ

Or, cos 2θ = 2 cos^{2} θ – 1

Or, 2 cos^{2} θ = cos 2θ + 1

Replace θ by θ/2.

Substituting the value of in

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