Proof:

To show that the expression 3(sinθ - cosθ)^4 + 6(sinθ cosθ)^2 + 4(sin^6θ cos^6θ) is independent of θ, we need to demonstrate that it does not change with different values of θ.

Expanding and Simplifying the Expression:

Let's start by expanding and simplifying the given expression step by step.

Step 1: Expand (sinθ - cosθ)^4

Using the binomial theorem, we can expand (sinθ - cosθ)^4 as follows:

(sinθ - cosθ)^4 = C(4,0)sin^4θ - C(4,1)sin^3θ cosθ + C(4,2)sin^2θ cos^2θ - C(4,3)sinθ cos^3θ + C(4,4)cos^4θ

Simplifying the terms:

= sin^4θ - 4sin^3θ cosθ + 6sin^2θ cos^2θ - 4sinθ cos^3θ + cos^4θ

Step 2: Expand (sinθ cosθ)^2

Expanding (sinθ cosθ)^2:

(sinθ cosθ)^2 = sin^2θ cos^2θ

Step 3: Simplify (sin^6θ cos^6θ)

We can simplify sin^6θ cos^6θ as follows:

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

Step 4: Substitute the expanded and simplified terms back into the expression

Substituting the expanded and simplified terms back into the original expression:

3(sinθ - cosθ)^4 + 6(sinθ cosθ)^2 + 4(sin^6θ cos^6θ)

= 3(sin^4θ - 4sin^3θ cosθ + 6sin^2θ cos^2θ - 4sinθ cos^3θ + cos^4θ) + 6(sin^2θ cos^2θ) + 4(sin^2θ cos^2θ)^3

Step 5: Simplify and Combine Like Terms

Simplifying further and combining like terms:

= 3sin^4θ - 12sin^3θ cosθ + 18sin^2θ cos^2θ - 12sinθ cos^3θ + 3cos^4θ + 6sin^2θ cos^2θ + 4(sin^2θ cos^2θ)^3

= 3sin^4θ + 6sin^2θ cos^2θ + 3cos^4θ - 12sin^3θ cosθ - 12sinθ cos^3θ + 4(sin^2θ cos^2θ)^3

Observing the Final Expression:

Upon careful observation of the final expression, we can see that all the terms involve powers of trigonometric functions (sinθ and cosθ) raised to even powers. This means that the expression is symmetric with respect to θ and will not change as

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