1. **State the problem:** We are given the function $f(x) = \frac{\cot x}{1 + \csc x}$ and need to simplify it. 2. **Recall the definitions:** $\cot x = \frac{\cos x}{\sin x}$ and $\csc x = \frac{1}{\sin x}$. 3. **Rewrite the function using these definitions:** $$f(x) = \frac{\frac{\cos x}{\sin x}}{1 + \frac{1}{\sin x}}$$ 4. **Simplify the denominator:** $$1 + \frac{1}{\sin x} = \frac{\sin x + 1}{\sin x}$$ 5. **Rewrite the function as:** $$f(x) = \frac{\frac{\cos x}{\sin x}}{\frac{\sin x + 1}{\sin x}} = \frac{\cos x}{\sin x} \times \frac{\sin x}{\sin x + 1}$$ 6. **Cancel $\sin x$ in numerator and denominator:** $$f(x) = \frac{\cos x}{\sin x + 1}$$ 7. **Final simplified form:** $$f(x) = \frac{\cos x}{\sin x + 1}$$ This is the simplified expression for the given function. **Explanation:** We used the fundamental trigonometric identities to rewrite the function and then simplified by combining fractions and canceling common terms. This approach helps in understanding how to manipulate trigonometric expressions effectively.