Let's break down the given expression step by step:
[tex]\[
\frac{1}{2} \left( \frac{a^2 + x^2}{a^2 - x^2} \right) - \frac{1}{2} \left( \frac{a + x}{a + x} \right) - \left( \frac{a}{a + x} \right)^2
\][/tex]
### Step 1: Simplify the second term
The second term [tex]\(\frac{1}{2} \left( \frac{a + x}{a + x} \right)\)[/tex] can be simplified because [tex]\(\frac{a + x}{a + x} = 1\)[/tex]. Hence, the term becomes:
[tex]\[
\frac{1}{2}
\][/tex]
### Step 2: Consider three separate terms
Label the terms for clarity as follows:
[tex]\[
\text{First term: } \frac{1}{2} \left( \frac{a^2 + x^2}{a^2 - x^2} \right)
\][/tex]
[tex]\[
\text{Second term: } \frac{1}{2}
\][/tex]
[tex]\[
\text{Third term: } \left( \frac{a}{a + x} \right)^{2}
\][/tex]
### Step 3: Combine the three terms
Now we express the given problem as a combination of these terms:
[tex]\[
\text{Expression} = \frac{1}{2} \left( \frac{a^2 + x^2}{a^2 - x^2} \right) - \frac{1}{2} - \left( \frac{a}{a + x} \right)^{2}
\][/tex]
### Step 4: Use the result of the respective expressions
Based on previous calculations, we can identify the numerical and simplified forms of these terms:
1. The first term [tex]\(\frac{1}{2} \left( \frac{a^2 + x^2}{a^2 - x^2} \right)\)[/tex] remains in its symbolic form:
[tex]\[
\frac{1}{2} \left( \frac{a^2 + x^2}{a^2 - x^2} \right)
\][/tex]
2. The second term, as simplified earlier:
[tex]\[
\frac{1}{2}
\][/tex]
3. The third term simplified to:
[tex]\[
\left( \frac{a}{a + x} \right)^{2}
\][/tex]
Now, combine these simplified forms and express the overall simplified result:
[tex]\[
\frac{1}{2} \left( \frac{a^2 + x^2}{a^2 - x^2} \right) - \frac{1}{2} - \left( \frac{a}{a + x} \right)^{2}
\][/tex]
After combining and further simplification, we get the final result for the whole expression:
[tex]\[
\frac{a^3 - a^2 x - a x^2 - x^3}{-a^3 - a^2 x + a x^2 + x^3}
\][/tex]
Therefore, the final, simplified version of the original expression is:
[tex]\[
\frac{a^3 - a^2 x - a x^2 - x^3}{-a^3 - a^2 x + a x^2 + x^3}
\][/tex]
