If y = cot^-1{(√(1 + sin x) + √ (1 - sin x))/(√(1 + sin x))}, show that dy/dx is independent of x.

Question

If y = cot^-1\(\Big\{\cfrac{\sqrt{1+sin\,\mathrm x}+\sqrt{1+sin\,\mathrm x}}{\sqrt{1+sin\,\mathrm x}-\sqrt{1-sin\,\mathrm x}}\Big\},\) show that \(\frac{dy}{d\mathrm x}\) is independent of x.

Answer 1

y =  cot^-1\(\Big\{\cfrac{\sqrt{1+sin\,\mathrm x}+\sqrt{1+sin\,\mathrm x}}{\sqrt{1+sin\,\mathrm x}-\sqrt{1-sin\,\mathrm x}}\Big\}\) 

Multiplying numerator and denominator

Differentiating w.r.t x, we get