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y= tan^(-1)((sqrt(1+x^2)+sqrt(1-x^2))/(s...

`y= tan^(-1)((sqrt(1+x^2)+sqrt(1-x^2))/(sqrt(1+x^2)-sqrt(1-x^2)))`, where `-1 < x < 1`, find `dy/dx`

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Putting `x^(2)= cos 2theta,` we get
`y=tan^(-1)((sqrt(1+cos theta)+sqrt(1-cos theta))/(sqrt(1+cost 2theta)-sqrt(1-cost 2theta)))`
`=tan^(-1)((sqrt(2cos^(2) theta)+sqrt(2sin^(2) theta))/(sqrt(2cos^(2)theta)-sqrt(2 sin^(2)theta)))`
`=tan^(-1)((cos theta+sintheta)/(cos theta-sin theta))`
`=tan^(-1)((1+tan theta)/(1- tan theta))`
`=tan^(-1)(tan (pi//4 +theta))`
`[{:(because, 0lt x^(2) lt1 rArr0 lt cos 2 theta lt 1),(or ,0 lt 2theta lt pi//2), (or , 0 lt theta lt pi//4) , (or, pi//4 lt pi//4 + theta lt pi//2):}]`
`=(pi)/(4)+theta`
`(pi)/(4)+(1)/(2)cos^(-1) x^(2)`
`therefore" "(dy)/(dx)=0-(1)/(2)xx(1)/(sqrt(1-(x^(2))^(2))).(d)/(dx)(x^(2))`
`-(1)/(2)xx(2x)/(sqrt(1-x^(4)))=(-x)/(sqrt(1-x^(4)))`
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