Concept:
Express the given expression in terms of \(\sin\theta\) and \(\cos\theta\), then use the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\) and the definition \(\cot\theta = \frac{\cos\theta}{\sin\theta}\) to simplify directly.
Step 1: Write the expression using only \(\sin\theta\) and \(\cos\theta\).
Given:
\[
E = \frac{\cos\theta}{1-\cos\theta} + \frac{\sec\theta}{1+\sec\theta}.
\]
Substitute \(\sec\theta = \frac{1}{\cos\theta}\):
\[
E = \frac{\cos\theta}{1-\cos\theta} + \frac{\frac{1}{\cos\theta}}{1+\frac{1}{\cos\theta}}.
\]
Simplify the second term by multiplying numerator and denominator by \(\cos\theta\):
\[
\frac{\frac{1}{\cos\theta}}{1+\frac{1}{\cos\theta}} = \frac{1}{\cos\theta+1}.
\]
So,
\[
E = \frac{\cos\theta}{1-\cos\theta} + \frac{1}{1+\cos\theta}.
\]
Step 2: Combine the fractions over a common denominator.
\[
E = \frac{\cos\theta(1+\cos\theta) + (1-\cos\theta)}{(1-\cos\theta)(1+\cos\theta)}.
\]
Expand numerator:
\[
\cos\theta + \cos^2\theta + 1 - \cos\theta = 1 + \cos^2\theta.
\]
Denominator simplifies via difference of squares:
\[
(1-\cos\theta)(1+\cos\theta) = 1 - \cos^2\theta = \sin^2\theta.
\]
Thus,
\[
E = \frac{1 + \cos^2\theta}{\sin^2\theta}.
\]
Step 3: Split the fraction and introduce \(\cot\theta\).
\[
E = \frac{1}{\sin^2\theta} + \frac{\cos^2\theta}{\sin^2\theta} = \csc^2\theta + \cot^2\theta.
\]
Recall the Pythagorean identity \(\csc^2\theta = 1 + \cot^2\theta\). Substituting:
\[
E = (1 + \cot^2\theta) + \cot^2\theta = 1 + 2\cot^2\theta.
\]
Step 4: Write the final answer.
\[
\boxed{1+\cot^2\theta}
\]