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The expression 4(cos⁶ theta - sin⁶ theta) simplifies to which of the following?

1. cos³(2 theta) + 3 cos(2 theta)
2. cos³(2 theta) - 3 cos(2 theta)
3. 4 cos³ theta - 3 cos theta
4. 3 cos(2 theta) - cos³(2 theta)

Correct answer: cos³(2 theta) + 3 cos(2 theta)

Solution

Writing it as a difference of cubes gives cos(2 theta)*(1 - cos² theta sin² theta), and replacing cos² theta sin² theta with (1 - cos²(2 theta))/4 leads to cos³(2 theta) + 3cos(2 theta) after multiplying by 4.

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