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Question: The sum of the solutions $x \in R$ of the equation $\frac{3 \cos 2x + \cos^3 2x}{\cos^6 x - \sin^6 x...

The sum of the solutions x∈Rx \in R of the equation 3cos⁡2x+cos⁡32xcos⁡6x−sin⁡6x=x3−x2+6\frac{3 \cos 2x + \cos^3 2x}{\cos^6 x - \sin^6 x} = x^3 - x^2 + 6 is:

A

1

B

0

C

-1

D

3

Answer

-1

Explanation

Solution

The LHS simplifies to 4. The equation becomes 4=x3−x2+64 = x^3 - x^2 + 6, which is x3−x2+2=0x^3 - x^2 + 2 = 0. Factoring gives (x+1)(x2−2x+2)=0(x+1)(x^2 - 2x + 2) = 0. The quadratic factor has no real roots. The only real solution is x=−1x = -1.