If `sinx=(2t)/(1+t^2)` , `tany=(2t)/(1-t^2)` , find `(dy)/(dx)` .

Explanation

We have given that
`sinx=(2t)/(1+t^2)`
, `tany=(2t)/(1-t^2)`
`x=sin ^{-1} frac{2 t}{1+t^{2}}`and `y=tan^{-1}frac{2 t}{1-t^{2}}`

Differentiate `x` with respect to `t`

`frac{d x}{d t}=frac{1}{sqrt{1-(frac{2 t}{1+t^{2}})^{2}}}
times frac{2(1+t^{2})-(2 t)(2 t)}{(1+t^{2})^{2}}`

`frac{d x}{d t}=frac{2}{1+t^{2}}`

Differentiate `y` with respect to `t`

`frac{d y}{d t}=frac{1}{1+(frac{2 t}{1+t^{2}})^{2}}
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