Certainly! Let's solve the equation [tex]\(\frac{a+b}{a-b} - \frac{a-b}{a+b} = \frac{4ab}{a^2 - b^2}\)[/tex].
To begin, note that [tex]\(a^2 - b^2\)[/tex] can be factored as [tex]\((a+b)(a-b)\)[/tex]. We'll use this fact in our simplifications.
Here's the step-by-step solution:
1. Left-hand side manipulation:
Start with the left-hand side (LHS) of the equation:
[tex]\[
\frac{a+b}{a-b} - \frac{a-b}{a+b}
\][/tex]
2. Common Denominator:
To subtract these fractions, we need a common denominator, which is [tex]\((a-b)(a+b)\)[/tex]:
[tex]\[
\frac{(a+b)^2}{(a-b)(a+b)} - \frac{(a-b)^2}{(a-b)(a+b)}
\][/tex]
This combines as a single fraction:
[tex]\[
\frac{(a+b)^2 - (a-b)^2}{(a-b)(a+b)}
\][/tex]
3. Expand the Numerator:
Next, expand the expressions in the numerator:
[tex]\[
(a+b)^2 = a^2 + 2ab + b^2
\][/tex]
[tex]\[
(a-b)^2 = a^2 - 2ab + b^2
\][/tex]
So,
[tex]\[
(a+b)^2 - (a-b)^2 = (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2)
\][/tex]
4. Simplify the Numerator:
Combine like terms:
[tex]\[
a^2 + 2ab + b^2 - a^2 + 2ab - b^2 = 4ab
\][/tex]
The numerator simplifies to [tex]\(4ab\)[/tex].
5. LHS Simplified:
Now the left-hand side is:
[tex]\[
\frac{4ab}{(a-b)(a+b)}
\][/tex]
6. Right-hand side:
The right-hand side (RHS) of the original equation is given as:
[tex]\[
\frac{4ab}{a^2 - b^2}
\][/tex]
7. Recognize the RHS Denominator:
Notice that [tex]\(a^2 - b^2 = (a+b)(a-b)\)[/tex].
So, the right-hand side becomes:
[tex]\[
\frac{4ab}{(a+b)(a-b)}
\][/tex]
8. Equality Verification:
Both LHS and RHS are now:
[tex]\[
\frac{4ab}{(a+b)(a-b)}
\][/tex]
Since the forms are identical, we have shown that:
[tex]\[
\frac{a+b}{a-b} - \frac{a-b}{a+b} = \frac{4ab}{a^2 - b^2}
\][/tex]
This equation holds true for all values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] where the denominators [tex]\((a-b)\)[/tex] and [tex]\((a+b)\)[/tex] are not zero, meaning [tex]\(a \neq b\)[/tex] and [tex]\(a \neq -b\)[/tex].
