If `y=sin[2t a n^(-1){sqrt((1-x)/(1+x))}],"f i n d"(dy)/(dx)`

Explanation

Here, `y=sin [2 tan ^{-1}(sqrt{frac{1-x}{1+x}})]`


Put `x=cos 2 theta`


We have, `y=sin [2 tan ^{-1}(sqrt{frac{1-cos 2 theta}{1+cos 2 theta}})]`


`=sin [2 tan ^{-1}(sqrt{frac{2 sin ^{2} theta}{2 cos ^{2} theta}})]`


`=sin [2 tan ^{-1} sqrt{tan ^{2} theta}]`


`=sin [2 tan ^{-1}(tan theta)]`


`=sin (2 theta)`


`=sin [2 times frac{1}{2} cos ^{-1} x][` Since, `x=cos 2 theta]`


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