Question

Differentiate \[\log \left( cosec x - \cot x \right)\] ?

Answer 1

\[\text{Let} y = \log \left( cosec x - \cot x \right)\]

\[\text{ Differentiate it with respect to x we get}, \]

\[\frac{d y}{d x} = \frac{d}{dx}\log \left( cosec x - \cot x \right)\]

\[ = \frac{1}{\left( cosec x - \cot x \right)}\frac{d}{dx}\left( cosec x - \cot x \right)\]

\[ = \frac{1}{\left( cosec x - \cot x \right)} \times \left( - cosec x \cot x + {cosec}^2 x \right)\]

\[ = \frac{ cosec x\left( cosec x - \cot x \right) }{\left( cosec x - \cot x \right)}\]

\[ = cosecx \]

\[So, \frac{d}{dx}\left\{ \log \left( cosec x - \cot x \right) \right\} = cosec x\]