Question

# $\frac{d\left[\mathrm{log}\left(\mathrm{tan}x\right)\right]}{dx}=$

A

$2sec2x$

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B

$2\mathrm{cos}ec2x$

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C

$sec2x$

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D

$\mathrm{cos}ec2x$

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Solution

## The correct option is B $2\mathrm{cos}ec2x$ Explanation for correct option:Let, $\mathrm{y}=\mathrm{log}\left(\mathrm{tanx}\right)$Then,$\begin{array}{rcl}& ⇒& \frac{dy}{dx}=\left(\frac{1}{\mathrm{tan}x}\right)×se{c}^{2}x\\ & =& \frac{\mathrm{cos}x}{\mathrm{sin}x}×\frac{1}{{\mathrm{cos}}^{2}x}\\ & =& \frac{2}{2\mathrm{sin}x\mathrm{cos}x}\\ & =& \frac{2}{\mathrm{sin}2x}\\ & =& 2\mathrm{cos}ec2x\end{array}$Hence, correct option is $\mathbf{\left(}\mathbf{B}\mathbf{\right)}$.

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