The expression 8(sin^6θ + cos^6θ) − 12(sin^4θ + cos^4θ) is simplified using trigonometric identities. What constant value does it equal for all real θ?

Introduction / Context:
This problem involves simplifying an expression with higher powers of sine and cosine. The expression 8(sin^6θ + cos^6θ) − 12(sin^4θ + cos^4θ) uses sixth and fourth powers, but by applying algebraic identities and the basic relation sin^2θ + cos^2θ = 1, it can be reduced to a constant value that does not depend on θ. Such tasks are classic in algebraic trigonometry simplification.

Given Data / Assumptions:

• Expression: 8(sin^6θ + cos^6θ) − 12(sin^4θ + cos^4θ).
• θ is any real angle.
• Identity: sin^2θ + cos^2θ = 1.
• Algebraic identities for sums of powers, especially a^3 + b^3 and a^2 + b^2.

Concept / Approach:
We express sin^6θ + cos^6θ and sin^4θ + cos^4θ in terms of sin^2θ and cos^2θ, and then in terms of their product sin^2θ cos^2θ. Once everything is written using u = sin^2θ cos^2θ, the expression turns into a linear expression in u, and the u terms cancel, leaving a constant.

Step-by-Step Solution:
Let s = sin^2θ and c = cos^2θ. Then s + c = 1. Compute sin^6θ + cos^6θ = s^3 + c^3. Use a^3 + b^3 = (a + b)^3 − 3ab(a + b). So s^3 + c^3 = (s + c)^3 − 3sc(s + c) = 1^3 − 3sc·1 = 1 − 3sc. Next compute sin^4θ + cos^4θ = s^2 + c^2. Use a^2 + b^2 = (a + b)^2 − 2ab. Thus s^2 + c^2 = (s + c)^2 − 2sc = 1^2 − 2sc = 1 − 2sc. Now substitute into the main expression: 8(sin^6θ + cos^6θ) − 12(sin^4θ + cos^4θ) = 8(1 − 3sc) − 12(1 − 2sc). Expand: 8 − 24sc − 12 + 24sc. Combine like terms: (8 − 12) + (−24sc + 24sc) = −4 + 0 = −4.
Verification / Alternative check:
You can check with a few random values of θ such as 0°, 30° or 45°. For example, at θ = 0°, sin θ = 0 and cos θ = 1, so the expression becomes 8·1 − 12·1 = −4, confirming the algebraic result.

Why Other Options Are Wrong:
Options 20 and −20 would require much larger magnitudes from the expression and do not match test values of θ. Option 4 has the correct magnitude but wrong sign. Option 0 would mean the expression vanishes for all θ, which is contradicted by direct substitution at θ = 0° or 90°.

Common Pitfalls:
A common mistake is to confuse sin^6θ with sin 6θ, which are completely different. Another pitfall is forgetting to use the basic identity s + c = 1, which greatly simplifies the algebra. Working systematically with s and c as sin^2θ and cos^2θ avoids these problems.

Final Answer:
The expression is identically equal to −4 for all real θ.