We are given the expression:
\[
\frac{\sin \theta(1 + \tan \theta) + \cos \theta(1 + \cot \theta)}{(\csc \theta - \sin \theta)(\sec \theta)(\cos \theta (\tan \theta + \cot \theta))}
\]
We begin by simplifying each term in the numerator and the denominator.
- Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
- Substituting these into the expression:
\[
\sin \theta (1 + \frac{\sin \theta}{\cos \theta}) + \cos \theta (1 + \frac{\cos \theta}{\sin \theta})
\]
This simplifies as follows:
\[
= \sin \theta \left(\frac{\cos \theta + \sin \theta}{\cos \theta}\right) + \cos \theta \left(\frac{\sin \theta + \cos \theta}{\sin \theta}\right)
\]
\[
= \frac{\sin \theta (\cos \theta + \sin \theta)}{\cos \theta} + \frac{\cos \theta (\sin \theta + \cos \theta)}{\sin \theta}
\]
Thus, the numerator simplifies to:
\[
\frac{\sin \theta (\cos \theta + \sin \theta)}{\cos \theta} + \frac{\cos \theta (\sin \theta + \cos \theta)}{\sin \theta}
\]
- The denominator involves the terms \( \csc \theta - \sin \theta \), \( \sec \theta \), and \( \tan \theta + \cot \theta \).
- Using the identities \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \), the expression becomes:
\[
(\frac{1}{\sin \theta} - \sin \theta) \cdot \frac{1}{\cos \theta} \cdot \cos \theta \cdot \left( \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} \right)
\]
This simplifies to:
\[
\frac{1 - \sin^2 \theta}{\sin \theta} \cdot \frac{1}{\cos \theta} \cdot \left(\frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}\right)
\]
- We now use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to further simplify the terms, and the expression simplifies to \( \csc \theta + \sec \theta \).
Thus, the value of the expression is:
\[
\boxed{\csc \theta + \sec \theta}
\]