Problem:

Evaluate the expression cos^3A sin^3A/cosAsinA cos^3A-sin^3A/cosA-sinA.

Solution:

To simplify the expression, let's break it down step by step.

Step 1:
Let's rewrite the expression using parentheses to make the order of operations clear:
(cos^3A sin^3A / cosAsinA) / (cos^3A - sin^3A / cosA - sinA)

Step 2:
Now, let's simplify the numerator and denominator separately.

Numerator:
In the numerator, we have cos^3A sin^3A / cosAsinA.

Key Point:
We can simplify this by canceling out common factors in the numerator and denominator.

Simplifying the numerator:
Using the quotient rule of exponents, we can rewrite cos^3A sin^3A as (cosA)^3 (sinA)^3.

Therefore, the numerator becomes (cosA)^3 (sinA)^3 / cosAsinA.

Key Point:
Notice that cosA is a factor in both the numerator and denominator. We can cancel it out.

Simplifying the numerator further:
After canceling out cosA, we are left with (sinA)^3 / sinA, which simplifies to sinA^2.

Denominator:
In the denominator, we have cos^3A - sin^3A / cosA - sinA.

Key Point:
We can simplify this using the difference of cubes formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2).

Simplifying the denominator:
Using the difference of cubes formula, we can rewrite cos^3A - sin^3A as (cosA - sinA)(cos^2A + cosA sinA + sin^2A).

Similarly, cosA - sinA can be written as (cosA - sinA)(1).

Therefore, the denominator becomes (cosA - sinA)(cos^2A + cosA sinA + sin^2A) / (cosA - sinA).

Step 3:
Now, let's simplify the entire expression, which is the numerator divided by the denominator:

(sinA)^2 / (cos^2A + cosA sinA + sin^2A).

Key Point:
We can simplify this expression by using the Pythagorean identity: sin^2A + cos^2A = 1.

Simplifying the expression:
Using the Pythagorean identity, we can rewrite sin^2A as 1 - cos^2A.

Therefore, the expression becomes (1 - cos^2A) / (cos^2A + cosA sinA + sin^2A).

Step 4:
Now, let's further simplify the expression by factoring.

Key Point:
We can factor the numerator and denominator to see if there are any common factors that can be canceled out.

Simplifying the expression further:
Factoring the numerator, we have (1 - cosA)(1 + cos