Step 1: Understanding the Concept:
To simplify trigonometric fractions, it is often best to convert all ratios into their basic forms, \(\sin \theta\) and \(\cos \theta\).
Step 2: Detailed Explanation:
The given expression is:
\[ E = \frac{\cos \theta}{1 - \tan \theta} + \frac{\sin \theta}{1 - \cot \theta} \]
Convert \(\tan \theta\) and \(\cot \theta\):
\[ E = \frac{\cos \theta}{1 - \frac{\sin \theta}{\cos \theta}} + \frac{\sin \theta}{1 - \frac{\cos \theta}{\sin \theta}} \]
Take LCM in the denominators:
\[ E = \frac{\cos \theta}{\frac{\cos \theta - \sin \theta}{\cos \theta}} + \frac{\sin \theta}{\frac{\sin \theta - \cos \theta}{\sin \theta}} \]
Invert and multiply:
\[ E = \frac{\cos^2 \theta}{\cos \theta - \sin \theta} + \frac{\sin^2 \theta}{\sin \theta - \cos \theta} \]
To make the denominators common, multiply the second term by -1:
\[ E = \frac{\cos^2 \theta}{\cos \theta - \sin \theta} - \frac{\sin^2 \theta}{\cos \theta - \sin \theta} \]
\[ E = \frac{\cos^2 \theta - \sin^2 \theta}{\cos \theta - \sin \theta} \]
Using identity \( a^2 - b^2 = (a-b)(a+b) \):
\[ E = \frac{(\cos \theta - \sin \theta)(\cos \theta + \sin \theta)}{\cos \theta - \sin \theta} \]
\[ E = \cos \theta + \sin \theta \].
Step 3: Final Answer:
The simplified expression is \( \sin \theta + \cos \theta \).