Question

If y=sin[2tan^-1((√1-x) ÷(1+x))],find dy÷dx

Answer 1

Answer:

L.H.S=\sin(2\tan^{1} \dfrac{\sqrt{1-x}}{\sqrt{1+x}})

Put x=\cos 2A

=\sin(2\tan^{1} \dfrac{\sqrt{1-\cos A}}{\sqrt{1+\cos A}})

=\sin(2\tan^{-1} \sqrt{{\dfrac{2\sin^2 A}{2\cos^2 A}}} )

=\sin(2\tan^{-1} \sqrt{{\dfrac{2\sin^2 A}{2\cos^2 A}}})

=\sin(2\tan^{-1} \sqrt{{\tan^2 A}})

=\sin 2A

∴ \sin 2A=\sqrt{1-\cos^2 2A}

\sin 2A=\sqrt{1-x^{2}}

=R.H.S, proved.

Hence, \sin(2\tan^{1} \dfrac{\sqrt{1-x}}{\sqrt{1+x}})=\sqrt{1-x^{2}}, proved.