Given:
$\frac{a}{b} \cdot \frac{b}{a} = -1$

To find:
$a^3 - b^3$

Solution:

Step 1: Simplifying the given equation

Given equation: $\frac{a}{b} \cdot \frac{b}{a} = -1$

We can simplify this equation by canceling out the common terms in the numerator and denominator:

$\frac{a}{b} \cdot \frac{b}{a} = \frac{ab}{ba} = 1$

Therefore, we have $1 = -1$.

Step 2: Understanding the equation

The equation $a^3 - b^3$ represents the difference of cubes, which can be factored as:

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

We need to find the value of $(a - b)(a^2 + ab + b^2)$.

Step 3: Using the given equation to find $(a - b)(a^2 + ab + b^2)$

From the given equation, we can rewrite it as:

$\frac{a}{b} \cdot \frac{b}{a} = \frac{ab}{ba} = 1$

Cross-multiplying, we get:

$ab = ba$

This implies that $a$ and $b$ are either equal or have opposite signs.

Case 1: $a = b$

If $a = b$, then $(a - b)(a^2 + ab + b^2) = 0$, since $(a - b) = 0$.

Case 2: $a$ and $b$ have opposite signs

If $a$ and $b$ have opposite signs, then $(a - b)$ will be negative. To find $(a - b)(a^2 + ab + b^2)$, we need to determine the sign of $(a^2 + ab + b^2)$.

We know that $(a - b)$ is negative, and since $(a - b)(a^2 + ab + b^2)$ equals $-1$, $(a^2 + ab + b^2)$ must be positive.

Therefore, in this case, $(a - b)(a^2 + ab + b^2) = -1$.

Step 4: Final answer

From the above analysis, we have two cases:

Case 1: $a = b$
In this case, $(a - b)(a^2 + ab + b^2) = 0$

Case 2: $a$ and $b$ have opposite signs
In this case, $(a - b)(a^2 + ab + b^2) = -1$

Hence, the expression $a^3 - b^3$ can either be 0 or -1, depending on the values of $a$ and $b$.