The value of _ e 2(d)/(dx)( _ x cosec x) at x = (π)/(4) is

For $\log_{e}2\dfrac{d}{dx}\left(\log_{\cos x} cosec x\right)$

let, $y = \log_{\cos x} cosec x$

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

$⇒ \dfrac{dy}{dx} = - \dfrac{\left[\cot x \cdot \ln \left(\cos x\right) + \tan x \cdot \ln \left(\sin x\right)\right]}{{\left(\ln \left(\cos x\right)\right)}^{2}}$

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

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

$⇒ \log_{e}2 \cdot \dfrac{4}{\ln 2} = 4$