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Question: If 0 \< x \<\(\frac{\pi}{2}\), y = cot<sup>–1</sup>\(\left\{ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \s...

If 0 < x <π2\frac{\pi}{2},

y = cot–1{1+sin⁡x+1−sin⁡x1+sin⁡x−1−sin⁡x}\left\{ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right\}, then dydx\frac{dy}{dx} =

A

–1

B

1

C

½

D

–1/2

Answer

½

Explanation

Solution

y = cot–1 1+sin⁡x+1−sin⁡x1+sin⁡x−1−sin⁡x\frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}}

= cot–1 {(cos⁡x2+sin⁡x2)2+(cos⁡x2−sin⁡x2)2(cos⁡x2+sin⁡x2)2−(cos⁡x2−sin⁡x2)2}\left\{ \frac{\sqrt{\left( \cos\frac{x}{2} + \sin\frac{x}{2} \right)^{2}} + \sqrt{\left( \cos\frac{x}{2} - \sin\frac{x}{2} \right)^{2}}}{\sqrt{\left( \cos\frac{x}{2} + \sin\frac{x}{2} \right)^{2} - \sqrt{\left( \cos\frac{x}{2} - \sin\frac{x}{2} \right)^{2}}}} \right\}

= cot–1 {cos⁡x2+sin⁡x2+cos⁡x2−sin⁡x2cos⁡x2+sin⁡x2−cos⁡x2+sin⁡x2}\left\{ \frac { \cos \frac { x } { 2 } + \sin \frac { x } { 2 } + \cos \frac { x } { 2 } - \sin \frac { x } { 2 } } { \cos \frac { x } { 2 } + \sin \frac { x } { 2 } - \cos \frac { x } { 2 } + \sin \frac { x } { 2 } } \right\} [⇒0<x2<π4]\left\lbrack \Rightarrow 0 < \frac{x}{2} < \frac{\pi}{4} \right\rbrack

= cot⁡−1(cot⁡x2)\cot^{- 1}\left( \cot\frac{x}{2} \right) = x2\frac{x}{2} ∴ dydx\frac{dy}{dx} = 12\frac{1}{2}