If frac{5pi}{2}

Question

(A) Given the expression $E = \frac{\sqrt{1 - \sin x} + \sqrt{1 + \sin x}}{\sqrt{1 - \sin x} - \sqrt{1 + \sin x}}$.Using $1 \pm \sin x = (\cos \frac{x

Vedclass · Accepted Answer

$-\cot \frac{x}{2}$

Vedclass · Answer

(A) Given the expression $E = \frac{\sqrt{1 - \sin x} + \sqrt{1 + \sin x}}{\sqrt{1 - \sin x} - \sqrt{1 + \sin x}}$.Using $1 \pm \sin x = (\cos \frac{x}{2} \pm \sin \frac{x}{2})^2$,we have $\sqrt{1 \pm \sin x} = |\cos \frac{x}{2} \pm \sin \frac{x}{2}|$.Since $\frac{5\pi}{2} In this interval,$\cos \frac{x}{2}  |\sin \frac{x}{2}|$.Thus,$\sqrt{1 - \sin x} = |\cos \frac{x}{2} - \sin \frac{x}{2}| = -(\cos \frac{x}{2} - \sin \frac{x}{2}) = \sin \frac{x}{2} - \cos \frac{x}{2}$.And $\sqrt{1 + \sin x} = |\cos \frac{x}{2} + \sin \frac{x}{2}| = -(\cos \frac{x}{2} + \sin \frac{x}{2}) = -\cos \frac{x}{2} - \sin \frac{x}{2}$.Substituting these into the expression:$E = \frac{(\sin \frac{x}{2} - \cos \frac{x}{2}) + (-\cos \frac{x}{2} - \sin \frac{x}{2})}{(\sin \frac{x}{2} - \cos \frac{x}{2}) - (-\cos \frac{x}{2} - \sin \frac{x}{2})} = \frac{-2\cos \frac{x}{2}}{2\sin \frac{x}{2}} = -\cot \frac{x}{2}$.

If $\frac{5\pi}{2} < x < 3\pi$,then the value of the expression $\frac{\sqrt{1 - \sin x} + \sqrt{1 + \sin x}}{\sqrt{1 - \sin x} - \sqrt{1 + \sin x}}$ is

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