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Question
\[\lim_{x \to \frac{\pi}{4}} \frac{2 - {cosec}^2 x}{1 - \cot x}\]
Solution
\[\lim_{x \to \frac{\pi}{4}} \left[ \frac{2 - {cosec}^2 x}{1 - \cot x} \right]\]
\[ = \lim_{x \to \frac{\pi}{4}} \left[ \frac{2 - \left( 1 + \cot^2 x \right)}{1 - \cot x} \right]\]
\[ = \lim_{x \to \frac{\pi}{4}} \left[ \frac{1 - \cot^2 x}{1 - \cot x} \right]\]
\[ = \lim_{x \to \frac{\pi}{4}} \left[ \frac{\left( 1 - \cot x \right) \left( 1 + \cot x \right)}{\left( 1 - \cot x \right)} \right]\]
\[ = 1 + \cot \left( \frac{\pi}{4} \right)\]
\[ = 1 + 1\]
\[ = 2\]
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