If , x² ≤ 1 ,then find   (CBSE 2015)

As the expression for y is quite complicated. So we need to simplify it before differentiation.

For simplification we can use idea like:

1 + cos 2θ = 2cos²θ or 1 - cos 2θ = 2sin²θ

For simplifying y, put x² = cos 2θ

∴ y =

⇒ y =

⇒ y =

Dividing numerator and denominator by cos θ

∴ y =

⇒ y =

∴ y =

Now differentiating y w.r.t x, we get –

∵ x² = cos 2θ ⇒ sin 2θ =

Differentiating the equation above w.r.t θ-

⇒

⇒

From above we have a value of sin 2θ

⇒

∴