If `y=cot^(-1){(sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))}.` Show that `(dy)/(dx)` is independent of `x.`

Explanation

Let, `{y}=cot ^{-1}[frac{sqrt{1+sin x}+sqrt{1-sin x}}{sqrt{1+sin x}-sqrt{1-sin x}}] ldots(i)`

We have,`frac{sqrt{1+sin x}+sqrt{1-sin x}}{sqrt{1+sin x}-sqrt{1-sin x}}`

`=frac{(sqrt{1+sin x}+sqrt{1-sin x})^{2}}{(sqrt{1+sin x}-sqrt{1-sin x})(sqrt{1+sin x}+sqrt{1-sin x})}`

`=frac{(1+sin x)+(1-sin x)+2 sqrt{(1-sin x)(1+sin x)}}{(1+sin x)-(1-sin x)}`

`=frac{2+2 sqrt{1-sin ^{2} x}}{2 sin x}`

`=frac{1+cos x}{sin x}` `=frac{2 cos ^{2} frac{x}{2}}{2 sin frac{x}{2} cos frac{x}{2}}`

`=cot frac{x}{2}`

Therefore, equation `(i)`

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