Sure! Let's break down the given mathematical expression step by step and build our final result.
Given the expression:
[tex]\[ 3[\sin x - \cos x]^4 + 6[\sin x + \cos x]^2 + 4[\sin^6 x + \cos^6 x] \][/tex]
### Step-by-Step Evaluation
1. Evaluate the first part: \( 3[\sin x - \cos x]^4 \)
- This expression takes the difference between \(\sin x\) and \(\cos x\), then raises it to the 4th power, and finally multiplies the result by 3.
2. Evaluate the second part: \( 6[\sin x + \cos x]^2 \)
- This expression takes the sum of \(\sin x\) and \(\cos x\), then squares it, and finally multiplies the result by 6.
3. Evaluate the third part: \( 4[\sin^6 x + \cos^6 x] \)
- This expression takes \(\sin x\) and \(\cos x\), raises each to the 6th power, adds these two results, and finally multiplies the sum by 4.
### Putting It All Together
Combining the results from the three parts mentioned above gives us:
[tex]\[
\boxed{3(\sin x - \cos x)^4 + 6(\sin x + \cos x)^2 + 4\sin^6 x + 4\cos^6 x}
\][/tex]
This is the final expression and represents the simplified form of the original problem.
