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Write the principal value of \( cot^{-1} \{ cot \bigg( -\frac{\pi}{4} \bigg) \} \)

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\[\frac{3\pi}{4}\]

Principal interval of cot is \((0.\pi)\)  and \(-\frac{\pi}{4}\notin\:(0,\pi)\)

We know that cot\(-\frac{\pi}{4}=cot(\pi-\frac{\pi}{4})=cot\frac{3\pi}{4}\)

\(\Rightarrow\:cot^{-1}cot-\frac{\pi}{4}=cot^{-1}cot\frac{3\pi}{4}=\frac{3\pi}{4}\)

which is in the principal interval of cot.
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