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If $y=\cot ^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right]$, where $0 < x < \frac{\pi}{2}$, then
$\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to
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$\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to
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Verified Answer
The correct answer is:
$\frac{1}{2}$
$\mathrm{y}=\cot ^{-1}\left[\frac{\sqrt{1+\sin \mathrm{x}}+\sqrt{1-\sin \mathrm{x}}}{\sqrt{1+\sin \mathrm{x}}-\sqrt{1-\sin \mathrm{x}}}\right]$
$\mathrm{y}=\cot ^{-1}$
$\left[\begin{array}{l}\sqrt{\cos ^{2} \frac{\mathrm{x}}{2}+\sin ^{2} \frac{\mathrm{x}}{2}+2 \sin \frac{\mathrm{x}}{2} \cos \frac{\mathrm{x}}{2}} \\ +\sqrt{\cos ^{2} \frac{\mathrm{x}}{2}+\sin ^{2} \frac{\mathrm{x}}{2}-2 \sin \frac{\mathrm{x}}{2} \cos \frac{\mathrm{x}}{2}}{\sqrt{\cos ^{2} \frac{\mathrm{x}}{2}+\sin ^{2} \frac{\mathrm{x}}{2}+2 \sin \frac{\mathrm{x}}{2} \cos \frac{\mathrm{x}}{2}}} \\ -\sqrt{\cos ^{2} \frac{\mathrm{x}}{2}+\sin ^{2} \frac{\mathrm{x}}{2}-2 \sin \frac{\mathrm{x}}{2} \cos \frac{\mathrm{x}}{2}}\end{array}\right]$
$y=\cot ^{-1}\left[\frac{\sqrt{\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)^{2}}+\sqrt{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)^{2}}}{\sqrt{\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)^{2}}-\sqrt{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)^{2}}}\right]$
$y=\cot ^{-1}\left[\frac{\cos \frac{x}{2}+\sin \frac{x}{2}+\cos \frac{x}{2}-\sin \frac{x}{2}}{\cos \frac{x}{2}+\sin \frac{x}{2}-\cos \frac{x}{2}+\sin \frac{x}{2}}\right]$
$y=\cot ^{-1}\left[\frac{2 \cos \frac{x}{2}}{2 \sin \frac{x}{2}}\right]=\cot ^{-1}\left(\cot \frac{x}{2}\right)=\frac{x}{2}$
$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2}$
$\mathrm{y}=\cot ^{-1}$
$\left[\begin{array}{l}\sqrt{\cos ^{2} \frac{\mathrm{x}}{2}+\sin ^{2} \frac{\mathrm{x}}{2}+2 \sin \frac{\mathrm{x}}{2} \cos \frac{\mathrm{x}}{2}} \\ +\sqrt{\cos ^{2} \frac{\mathrm{x}}{2}+\sin ^{2} \frac{\mathrm{x}}{2}-2 \sin \frac{\mathrm{x}}{2} \cos \frac{\mathrm{x}}{2}}{\sqrt{\cos ^{2} \frac{\mathrm{x}}{2}+\sin ^{2} \frac{\mathrm{x}}{2}+2 \sin \frac{\mathrm{x}}{2} \cos \frac{\mathrm{x}}{2}}} \\ -\sqrt{\cos ^{2} \frac{\mathrm{x}}{2}+\sin ^{2} \frac{\mathrm{x}}{2}-2 \sin \frac{\mathrm{x}}{2} \cos \frac{\mathrm{x}}{2}}\end{array}\right]$
$y=\cot ^{-1}\left[\frac{\sqrt{\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)^{2}}+\sqrt{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)^{2}}}{\sqrt{\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)^{2}}-\sqrt{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)^{2}}}\right]$
$y=\cot ^{-1}\left[\frac{\cos \frac{x}{2}+\sin \frac{x}{2}+\cos \frac{x}{2}-\sin \frac{x}{2}}{\cos \frac{x}{2}+\sin \frac{x}{2}-\cos \frac{x}{2}+\sin \frac{x}{2}}\right]$
$y=\cot ^{-1}\left[\frac{2 \cos \frac{x}{2}}{2 \sin \frac{x}{2}}\right]=\cot ^{-1}\left(\cot \frac{x}{2}\right)=\frac{x}{2}$
$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2}$
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