Dear Student, put x2=cos 2θ=sin-11+cos 2θ2=sin-12cos2θ2=sin-1(cos θ)=sin-1sin π2-θ=π2-θy=π2-cos-1x22derivative with respect to x dydx=-12-11-x4×2xdydx=x1-x4 Regards!

Find the derivative.

Dear Student,

p u t x^{2} = \cos 2 θ = \sin^{- 1} \sqrt{\frac{1 + \cos 2 θ}{2}} = \sin^{- 1} \sqrt{\frac{2 \cos^{2} θ}{2}} = \sin^{- 1} (\cos θ) = \sin^{- 1} \{\sin (\frac{π}{2} - θ)\} = \frac{π}{2} - θ y = \frac{π}{2} - \frac{\cos^{- 1} x^{2}}{2} d e r i v a t i v e w i t h r e s p e c t t o x . \frac{d y}{d x} = - \frac{1}{2} \frac{- 1}{\sqrt{1 - x^{4}}} \times 2 x \frac{d y}{d x} = \frac{x}{\sqrt{1 - x^{4}}}

Regards!

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