Let's solve the given complex fraction step by step to determine which expression it is equivalent to.
We start with the complex fraction:
[tex]\[
\frac{\frac{1}{x} - \frac{1}{y}}{\frac{1}{x} + \frac{1}{y}}
\][/tex]
To simplify this, we first resolve the numerator and the denominator separately and then simplify the resulting expression.
### Step 1: Simplify the Numerator
The numerator is:
[tex]\[
\frac{1}{x} - \frac{1}{y}
\][/tex]
To combine these two fractions into a single fraction, find a common denominator, which is [tex]\( xy \)[/tex]:
[tex]\[
\frac{1}{x} - \frac{1}{y} = \frac{y}{xy} - \frac{x}{xy} = \frac{y - x}{xy}
\][/tex]
### Step 2: Simplify the Denominator
The denominator is:
[tex]\[
\frac{1}{x} + \frac{1}{y}
\][/tex]
Similarly, find a common denominator of [tex]\( xy \)[/tex]:
[tex]\[
\frac{1}{x} + \frac{1}{y} = \frac{y}{xy} + \frac{x}{xy} = \frac{y + x}{xy}
\][/tex]
### Step 3: Form the Complex Fraction
Now, substitute the simplified forms of the numerator and the denominator back into the complex fraction:
[tex]\[
\frac{\frac{1}{x} - \frac{1}{y}}{\frac{1}{x} + \frac{1}{y}} = \frac{\frac{y - x}{xy}}{\frac{y + x}{xy}}
\][/tex]
### Step 4: Simplify the Complex Fraction
When dividing two fractions, you multiply by the reciprocal of the denominator:
[tex]\[
\frac{\frac{y - x}{xy}}{\frac{y + x}{xy}} = \frac{y - x}{xy} \times \frac{xy}{y + x} = \frac{y - x}{y + x}
\][/tex]
### Conclusion
The simplified form of the given complex fraction is [tex]\( \frac{y - x}{y + x} \)[/tex].
Comparing this with the given options, we see that it matches option 4:
\[
\boxed{\frac{y - x}{y + x}}
