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Question
\[\lim_{x \to 0} \left( cosec x - \cot x \right)\]
Solution
\[\lim_{x \to 0} \left[ cosec x - \cot x \right]\]
\[ = \lim_{x \to 0} \left[ \frac{1}{\sin x} - \frac{\cos x}{\sin x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{1 - \cos x}{\sin x} \right] \left[ \because 1 - \cos A = 2 \sin^2 \left( \frac{A}{2} \right) \right]\]
\[ = \lim_{x \to 0} \left[ \frac{2 \sin^2 \left( \frac{x}{2} \right)}{2 \sin \frac{x}{2} \cos \frac{x}{2}} \right]\]
\[ = \lim_{x \to 0} \left( \tan \left( \frac{x}{2} \right) \right)\]
\[ = 0\]
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