To prove the equation:

sin^4(theta) * cos^4(theta) / (1 - 2sin^2(theta) * cos^2(theta)) = 1

We will start by manipulating the left-hand side of the equation using trigonometric identities.

1. Simplifying the denominator:
1 - 2sin^2(theta) * cos^2(theta)
Using the double angle identity for sine: sin(2theta) = 2sin(theta) * cos(theta)
We can rewrite the denominator as: 1 - sin^2(2theta)
Applying the Pythagorean identity: sin^2(x) + cos^2(x) = 1
We have: 1 - sin^2(2theta) = cos^2(2theta)

2. Rewriting the numerator:
sin^4(theta) * cos^4(theta)
Applying the identity: sin^2(x) = 1 - cos^2(x)
We can rewrite the numerator as: (1 - cos^2(theta))^2 * cos^4(theta)
Expanding the square: (1 - 2cos^2(theta) + cos^4(theta)) * cos^4(theta)
Simplifying: cos^4(theta) - 2cos^6(theta) + cos^8(theta)

3. Substituting the rewritten numerator and denominator back into the equation:
(cos^4(theta) - 2cos^6(theta) + cos^8(theta)) / cos^2(2theta) = 1

4. Canceling out common terms:
Dividing the numerator by cos^2(2theta):
cos^4(theta) - 2cos^6(theta) + cos^8(theta) = cos^2(2theta)

5. Rearranging terms:
cos^8(theta) - 2cos^6(theta) + cos^4(theta) - cos^2(2theta) = 0

6. Factoring the equation:
(cos^4(theta) - cos^2(2theta))(cos^4(theta) - 2cos^4(theta) + 1) = 0

7. Simplifying further:
cos^4(theta) - cos^2(2theta) = 0 (Equation 1)
cos^4(theta) - 2cos^4(theta) + 1 = 0 (Equation 2)

Now, we need to prove that both Equation 1 and Equation 2 hold true.

For Equation 1:
Using the double angle identity for cosine: cos(2theta) = 1 - 2sin^2(theta)
We can rewrite Equation 1 as: cos^4(theta) - (1 - 2sin^2(theta))^2 = 0
Expanding the square: cos^4(theta) - (1 - 4sin^2(theta) + 4sin^4(theta)) = 0
Simplifying: cos^4(theta) - 1 + 4sin^2(theta) - 4sin^4(theta) = 0
Rearranging terms: -4sin^4(theta) + 4sin^2(theta) + cos^4(theta) - 1 = 0
Using the Pythagorean identity: sin^2(theta