If tanA and tanB are the roots of the quadratic equation, 3x² - x - 5 = 0, then the value of 3sin²(A+B) - sin(A+B).cos(A+B) - cos²(A+B) is?

Using the fact that tanA and tanB are the roots of 3x² - x - 5 = 0,
Sum of the roots = 1/3
Product of the roots = -5/3
tan(A+B) = (tanA + tanB) / (1 - tanA.tanB)
we get , tan(A+B) = (1/3) / (1 - (-5/3)) = 1/3 / (8/3) = 1/8
We have, cos²(A+B) = 1 / (1 + tan²(A+B)) = 1 / (1 + (1/8)²) = 64/65
We see that 3sin²(A+B) - sin(A+B)cos(A+B) - cos²(A+B) = cos²(A+B)(3tan²(A+B) - tan(A+B) - 1) = (64/65)(3(1/64) - (1/8) - 1) = (64/65)(3/64 - 8/64 - 64/64) = (64/65)(-69/64) = -69/65
So option C is the correct answer.