Unique limits of integration

Calculus Level 4

$\large \displaystyle \int \limits_{\arccos\left( \ln \left(\pi/4 \right) \right)}^{\arccos \left( \ln \left(\pi/2 \right) \right)} (-\sin x)e^{\cos x} \cot \left( e^{\cos x}\right) \, dx$

If the above integral can be represented in the form $$\ln \sqrt{a}$$, find $$a$$.

Clarification: $$e \approx 2.71828$$ denotes the Euler's number.

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