In this post, I will show a step-by-step approach to prove that \(\displaystyle \frac{d}{dx}\csc(x) = -\csc(x) \cot(x) \)
Proof:
\(
\displaystyle
\frac{d}{dx}\csc(x) \\
\displaystyle
= \frac{d}{dx}\left(\frac{1}{\sin(x)}\right) \\
\displaystyle
= \frac{-\cos(x)}{\sin^2(x)} \scriptsize \text{, using quotient rule} \\
\displaystyle
= \frac{-1}{\sin(x)}\frac{\cos(x)}{\sin(x)} \\
\displaystyle
= -\csc(x) \cot(x) \scriptsize \text{, since } \frac{\cos(x)}{\sin(x)} = \cot(x) \text{ and } \frac{1}{\sin(x)} = \csc(x) \\ \\
\text{Hence, proved: } \fbox{$\frac{d}{dx} \csc(x) = -\csc(x) \cot(x)$}
\)
