Expression:
The given expression is sin^6 - cos^6.

Solution:
To evaluate the given expression sin^6 - cos^6, we can use the trigonometric identity:

sin^2θ - cos^2θ = 1

We can rewrite the expression as:

sin^6 - cos^6 = (sin^2)^3 - (cos^2)^3

Using the identity sin^2θ - cos^2θ = 1, we can substitute:

(sin^2)^3 - (cos^2)^3 = (1 - cos^2)^3 - (cos^2)^3

Step 1: Expand the expression:

(1 - cos^2)^3 - (cos^2)^3 = (1 - 3cos^2 + 3cos^4 - cos^6) - (cos^6)

Simplifying further:

= 1 - 3cos^2 + 3cos^4 - cos^6 - cos^6

Step 2: Combine like terms:

= 1 - 3cos^2 - 2cos^6 + 3cos^4

Step 3: Rearrange the terms:

= 1 - 2cos^6 - 3cos^2 + 3cos^4

Step 4: Factor out the common terms:

= 1 - cos^2(2cos^4 + 3 - 3cos^2)

Step 5: Simplify the expression inside the parentheses:

= 1 - cos^2(2cos^4 - 3cos^2 + 3)

Step 6: Evaluate the expression inside the parentheses:

Let's substitute x = cos^2:

= 1 - x(2x^2 - 3x + 3)

Step 7: Simplify the expression inside the parentheses:

= 1 - 2x^3 + 3x^2 - 3x

Step 8: Substitute back cos^2 for x:

= 1 - 2(cos^2)^3 + 3(cos^2)^2 - 3(cos^2)

Step 9: Simplify further:

= 1 - 2cos^6 + 3cos^4 - 3cos^2

Therefore, the value of sin^6 - cos^6 is 1 - 2cos^6 + 3cos^4 - 3cos^2.