If , then prove that .

OR

If x = a(cos2θ + 2θsin2θ) and y = a(sin2θ – 2θcos2θ), find at θ = .

Given:

To Prove:

Let x = sinA and y = sinB ---- (i)

Putting the value in the equation we get,

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

Now putting the values of A and B from (i),

⇒ sin^-1x – sin^-1y = 2cot^-1a

Now differentiating with respect to x we get,

⇒

⇒

⇒ [Proved]

OR

Given: x = a(cos2θ + 2θsin2θ) and y = a(sin2θ – 2θcos2θ)

To Find: at θ =

Differentiating x with respect to θ we get,

-- (i)

Differentiating y with respect to θ we get,

From the above two equations we get,

⇒

Differentiating with respect to x we get,

⇒

⇒ [Putting the value of from (i)]

At θ = ,

⇒