Differentiate the following functions with respect to x: (cosec x… - Vidyadip

Question

Differentiate the following functions with respect to x:
$\log(\text{cosec x}-\cot\text{x})$

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Answer

Consider $\text{y}=\log(\text{cosec x}-\cot\text{x})$
Differentiate with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\log\big(\text{cosec x}-\cot\text{x}\big)$
$=\frac{1}{(\text{cosec x}-\cot\text{x})}\times\big(-\text{cosec x}\cot\text{x}+\text{cosec}^2\text{x}\big)$
[Using chain rule]
$=\frac{1}{(\text{cosec x}-\cot\text{x})}\times\big(-\text{cosec x}\cot\text{x}+\text{cosec}^2\text{x}\big)$
$=\frac{\text{cosec x}(\text{cosec x}-\cos\text{x})}{(\text{cosec x}-\cot\text{x})}$
$=\text{cosec x}$
Hence, the solution is, $\frac{\text{d}}{\text{dx}}\big(\log(\text{cosec x}-\cot\text{x})\big)=\text{cosec x}$

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