The function provided is:

$f(x) = \frac{\cot x}{1 + \csc x}$

Steps to Simplify or Analyze

1. Rewrite in terms of sine and cosine:

   • Recall that $\cot x = \frac{\cos x}{\sin x}$ and $\csc x = \frac{1}{\sin x}$.
   • Substituting these identities, we get: $f(x) = \frac{\frac{\cos x}{\sin x}}{1 + \frac{1}{\sin x}}$

2. Simplify the denominator:

   • The denominator $1 + \frac{1}{\sin x}$ becomes: $\frac{\sin x + 1}{\sin x}$

3. Simplify the fraction:

   • The function now becomes: $f(x) = \frac{\cos x}{\sin x} \cdot \frac{\sin x}{\sin x + 1}$
   • Simplify: $f(x) = \frac{\cos x}{\sin x + 1}$

Final Simplified Form:

$f(x) = \frac{\cos x}{\sin x + 1}$

Notes:

• The function is undefined at points where $\sin x + 1 = 0$, i.e., $\sin x = -1$. This occurs at $x = 3\pi/2 + 2n\pi$, where $n \in \mathbb{Z}$.
• Additionally, the function is undefined where $\sin x = 0$ because of the original $\cot x$ term, i.e., at $x = n\pi$, where $n \in \mathbb{Z}$.

Would you like me to:

1. Discuss the derivative of this function?
2. Solve for specific values of $x$?
3. Explore the domain and range in detail?
4. Graph the simplified function?
5. Analyze asymptotes or critical points?

Tip:

Always check for undefined points when dealing with trigonometric functions to avoid domain-related issues!