Unique limits of integration

Put $\large e^{\cos x}=u$

$\implies$ $\large e^{\cos x}(-\sin x)\mathrm{d}x=\mathrm{d}u$

and lower and upper limits of integration are

$\color{#D61F06}{\Large e^{\cos (\color{#20A900}{\cos^{-1}\left( \ln \left(\pi/4 \right) \right)})}}$

and $\large \color{#3D99F6}{\Large e^{\cos(\color{#20A900}{\cos^{-1}\left( \ln \left(\pi/2 \right) \right)})}}$

i.e $\large\color{#D61F06}{\dfrac{\pi}{4}}$ and $\large\color{#3D99F6}{\dfrac{\pi}{2}}$ respectively.

$\Large\mathfrak{T}=\displaystyle\int_{\color{#D61F06}{\pi/4}}^{\color{#3D99F6}{\pi/2}}\cot u ~\mathrm{d}u$

$\LARGE =\left|\ln(\sin u)\right|_{\color{#D61F06}{\pi/4}}^{\color{#3D99F6}{\pi/2}}$ $\LARGE =\ln(\sqrt 2)$

$\Huge \therefore \boxed{\color{#69047E}{a=2}}$