Express the following in simplest form: (complex fraction) Sin(x) - Cos(x) Cos(x) Sin(x) _______________ 1 - 1 Cos(x) Sin(x)

To simplify the given complex fraction, we can simplify the numerator and the denominator separately and then simplify the entire fraction.

Let's start by simplifying the numerator: Sin(x) - Cos(x) Cos(x) Sin(x).

We can rewrite the numerator as follows:
Sin(x) - (Cos(x))^2 * (Sin(x))

Using the trigonometric identity, (Cos(x))^2 = 1 - (Sin(x))^2, we can substitute in the numerator:
Sin(x) - (1 - (Sin(x))^2) * Sin(x)

Expanding the multiplication:
Sin(x) - Sin(x) + (Sin(x))^3

Now, we simplify further:
(Sin(x))^3

Moving on to the denominator: 1 - 1 * Cos(x) * Sin(x).

We can rewrite the denominator as:
1 - Cos(x) * Sin(x)

Next, we need to simplify the entire fraction by dividing the simplified numerator by the simplified denominator:

(Sin(x))^3
_____________
1 - Cos(x) * Sin(x)

And there you have it, the given complex fraction, (Sin(x) - Cos(x) Cos(x) Sin(x)) / (1 - 1 Cos(x) Sin(x)), is simplified to (Sin(x))^3 / (1 - Cos(x) Sin(x)).