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# Solution for If Y = Cot − 1 { √ 1 + Sin X + √ 1 − Sin X √ 1 + Sin X − √ 1 − Sin X } , Show that D Y D X is Independent of X. ? - CBSE (Commerce) Class 12 - Mathematics

ConceptSimple Problems on Applications of Derivatives

#### Question

If  $y = \cot^{- 1} \left\{ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right\}$,  show that $\frac{dy}{dx}$ is independent of x. ?

#### Solution

$\text{ Let, y } = co t^{- 1} \left[ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right] . . . \left( i \right)$
$\text{We have }, \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}}$
$= \frac{\left( \sqrt{1 + \sin x} + \sqrt{1 - \sin x} \right)^2}{\left( \sqrt{1 + \sin x} - \sqrt{1 - \sin x} \right)\left( \sqrt{1 + \sin x} + \sqrt{1 - \sin x} \right)}$
$= \frac{\left( 1 + \sin x \right) + \left( 1 - \sin x \right) + 2\sqrt{\left( 1 - \sin x \right)\left( 1 + \sin x \right)}}{\left( 1 + \sin x \right) - \left( 1 - \sin x \right)}$
$= \frac{2 + 2\sqrt{1 - \sin^2 x}}{2\sin x}$
$= \frac{1 + \cos x}{\sin x}$
$= \frac{2 \cos^2 \frac{x}{2}}{2\sin\frac{x}{2}\cos\frac{x}{2}}$
$= cot\frac{x}{2}$
$\text{ Therefore, equation } \left( i \right) \text{ becomes}$
$y = co t^{- 1} \left( cot\frac{x}{2} \right)$
$\Rightarrow y = \frac{x}{2}$
$\therefore \frac{d y}{d x} = \frac{1}{2}$
$\text{ Hence }, \frac{d y}{d x} \text{ is independent of x} .$

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Solution If Y = Cot − 1 { √ 1 + Sin X + √ 1 − Sin X √ 1 + Sin X − √ 1 − Sin X } , Show that D Y D X is Independent of X. ? Concept: Simple Problems on Applications of Derivatives.
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