The value of log e 2 d dx log cos x cosec x at x = π 4 is

For ${\log }_{\mathrm{e}}2\frac{d}{dx}\left({\log }_{\cos x}cosecx\right)$

let, $y={\log }_{\cos x}cosecx$

i.e. $y=-\frac{\ln \left(\sin x\right)}{\ln \left(\cos x\right)}$

$\Rightarrow \frac{dy}{dx}=-\frac{\left[\cot x·\ln \left(\cos x\right)+\tan x·\ln \left(\sin x\right)\right]}{{\left(\ln \left(\cos x\right)\right)}^2}$

Now ${\left(\frac{dy}{dx}\right)}_{x=\frac{\pi }{4}}=-\frac{\left[\cot \frac{\pi }{4}·\ln \left(\cos \frac{\pi }{4}\right)+\tan \frac{\pi }{4}·\ln \left(\sin \frac{\pi }{4}\right)\right]}{{\left(\ln \left(\cos \frac{\pi }{4}\right)\right)}^2}$

$=\frac{4}{\ln 2}$

$\Rightarrow {\log }_{e}2·\frac{4}{\ln 2}=4$