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

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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