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Question
Differentiate \[\log \left( cosec x - \cot x \right)\] ?
Solution
\[\text{Let} y = \log \left( cosec x - \cot x \right)\]
\[\text{ Differentiate it with respect to x we get}, \]
\[\frac{d y}{d x} = \frac{d}{dx}\log \left( cosec x - \cot x \right)\]
\[ = \frac{1}{\left( cosec x - \cot x \right)}\frac{d}{dx}\left( cosec x - \cot x \right)\]
\[ = \frac{1}{\left( cosec x - \cot x \right)} \times \left( - cosec x \cot x + {cosec}^2 x \right)\]
\[ = \frac{ cosec x\left( cosec x - \cot x \right) }{\left( cosec x - \cot x \right)}\]
\[ = cosecx \]
\[So, \frac{d}{dx}\left\{ \log \left( cosec x - \cot x \right) \right\} = cosec x\]
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