The function you've written appears to be:

$f(x) = \ln\left(\frac{1 + \sin(x)}{1 - \sin(x)}\right)$

This expression involves a logarithmic function where the argument is a fraction. Let's break it down:

Simplifying and Understanding the Function

The function is:

$f(x) = \ln\left(\frac{1 + \sin(x)}{1 - \sin(x)}\right)$

This is a logarithm with a ratio of two terms, $1 + \sin(x)$ and $1 - \sin(x)$. This function may have certain properties such as domain restrictions due to the presence of the sine function and the logarithm.

Domain of $f(x)$:

For the logarithm function to be defined, the argument must be strictly positive:

$\frac{1 + \sin(x)}{1 - \sin(x)} > 0$

This means that $1 + \sin(x)$ and $1 - \sin(x)$ must have the same sign (both positive or both negative).

• $1 + \sin(x) \geq 0$ for all $x$, because the sine function oscillates between -1 and 1, so $1 + \sin(x)$ is always non-negative.
• However, $1 - \sin(x)$ will be positive when $\sin(x) < 1$ and will approach zero as $\sin(x)$ approaches 1.

Thus, there will be specific intervals where the logarithmic function is valid, depending on the sine value.

Properties of $f(x)$:

To understand its behavior better, let's try some specific values of $x$:

1. When $x = 0$, $\sin(0) = 0$: $f(0) = \ln\left(\frac{1 + 0}{1 - 0}\right) = \ln(1) = 0$

2. When $x = \frac{\pi}{2}$, $\sin\left(\frac{\pi}{2}\right) = 1$: $f\left(\frac{\pi}{2}\right) = \ln\left(\frac{1 + 1}{1 - 1}\right)$ This expression is undefined because the denominator becomes zero.

The function is undefined where $1 - \sin(x) = 0$, which occurs when $\sin(x) = 1$, i.e., at $x = \frac{\pi}{2} + 2k\pi$, where $k$ is an integer.

Summary:

The function $f(x) = \ln\left(\frac{1 + \sin(x)}{1 - \sin(x)}\right)$ is well-defined for values of $x$ where $\sin(x) \neq 1$, i.e., excluding the points where $x = \frac{\pi}{2} + 2k\pi$, for any integer $k$.

Would you like to explore the function further, such as its derivative, integral, or graph? Feel free to ask for more details! Here are some related questions:

1. How do we find the derivative of $f(x)$?
2. How can we analyze the behavior of this function near the points where it is undefined?
3. How can we compute the integral of $f(x)$?
4. What are the asymptotic behaviors of the function near the undefined points?
5. How can we graph $f(x)$ to visualize its properties?

Tip: When dealing with logarithmic functions that involve trigonometric terms, it’s important to check the domain carefully to avoid undefined expressions.