The value of cot−1⁡1−sin⁡x+1+sin⁡x1−sin⁡x−1+sin⁡x is

# The value of ${\mathrm{cot}}^{-1}\left[\frac{\sqrt{1-\mathrm{sin}\mathrm{x}}+\sqrt{1+\mathrm{sin}\mathrm{x}}}{\sqrt{1-\mathrm{sin}\mathrm{x}}-\sqrt{1+\mathrm{sin}\mathrm{x}}}\right]$ is

1. A

$\mathrm{\pi }-\mathrm{x}$

2. B

$\mathrm{\pi }-\frac{\mathrm{x}}{2}$

3. C

$\mathrm{\pi }-\frac{\mathrm{x}}{3}$

4. D

None of these

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### Solution:

$\begin{array}{l}{\mathrm{cot}}^{-1}\left[\frac{\left(\mathrm{cos}\frac{1}{2}\mathrm{x}-\mathrm{sin}\frac{1}{2}\mathrm{x}\right)+\left(\mathrm{cos}\frac{1}{2}\mathrm{x}+\mathrm{sin}\frac{1}{2}\mathrm{x}\right)}{\left(\mathrm{cos}\frac{1}{2}\mathrm{x}-\mathrm{sin}\frac{1}{2}\mathrm{x}\right)-\left(\mathrm{cos}\frac{1}{2}\mathrm{x}+\mathrm{sin}\frac{1}{2}\mathrm{x}\right)}\right]\\ ={\mathrm{cot}}^{-1}\left(-\mathrm{cot}\frac{1}{2}\mathrm{x}\right)\\ ={\mathrm{cot}}^{-1}\mathrm{cot}\left(\mathrm{\pi }-\frac{1}{2}\mathrm{x}\right)=\mathrm{\pi }-\frac{1}{2}\mathrm{x}\end{array}$

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