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Let the domain and range of inverse circular functions are
\( \mathrm{P} \)
defined as follows
\begin{tabular}{|l|l|l|}
\hline & Domain & Range \\
\hline \( \sin ^{-1} x \) & {\( [-1,1] \)} & {\( \left[\frac{\pi}{2}, \frac{3 \pi}{2}\right] \)} \\
\hline \( \cos ^{-1} x \) & {\( [-1,1] \)} & {\( [0, \pi] \)} \\
\hline \( \tan ^{-1} x \) & \( R \) & \( \left(\frac{\pi}{2}, \frac{3 \pi}{2}\right) \) \\
\hline \( \cot ^{-1} x \) & \( R \) & \( (0, \pi) \) \\
\hline \( \operatorname{cosec}^{-1} x \) & \( (-\infty,-1] \cup[1, \infty) \) & {\( \left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]-\{\pi\} \)} \\
\hline \( \sec ^{-1} x \) & \( (-\infty,-1] \cup[1, \infty) \) & {\( [0, \pi]-\left\{\frac{\pi}{2}\right\} \)} \\
\hline
\end{tabular}

If \( x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)-\{0\} \), then \( \operatorname{cosec}^{-1}(\operatorname{cosec} x) \) is
(A) \( 2 \pi-x \)
(B) \( \pi+x \)
(C) \( \pi-x \)
(D) \( -\pi-x \)


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