Step 1: Analyze the Given Equation.
We are given:
\[
\frac{a}{b} + \frac{b}{a} = 1
\]
Rewrite this using a common denominator:
\[
\frac{a^2 + b^2}{ab} = 1
\]
Multiply both sides by \( ab \) (assuming \( a, b \neq 0 \)):
\[
a^2 + b^2 = ab
\]
Step 2: Use the Identity for \( a^3 + b^3 \).
The identity is:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
From earlier, we know:
\[
a^2 + b^2 = ab
\]
Now substitute into the expression \( a^2 - ab + b^2 \):
\[
a^2 - ab + b^2 = (a^2 + b^2) - ab = ab - ab = 0
\]
Thus:
\[
a^3 + b^3 = (a + b)(0) = 0
\]
Step 3: Analyze the Options.
Option (1): \( 1 \) — Incorrect
Option (2): \( -1 \) — Incorrect
Option (3): \( \frac{1}{2} \) — Incorrect
Option (4): \( 0 \) — Correct
Step 4: Final Answer.
\[
(4) \quad \mathbf{0}
\]