If \tan\alpha and \tan\beta are the roots of the equation x^{2}-6x+8=0, then what is the value of \cos(2\alpha+2\beta)?
- A. \frac{13}{75}
- B. \frac{13}{85} ✓
- C. \frac{17}{85}
- D. \frac{19}{85}
Correct Answer: B. \frac{13}{85}
Explanation
From the quadratic equation, sum of roots is \tan \alpha + \tan \beta = 6 and product is \tan \alpha \tan \beta = 8. Then, \tan(\alpha+\beta) = \frac{6}{1-8} = -\frac{6}{7}. Using the double angle formula, \cos(2(\alpha+\beta)) = \frac{1-\tan^2(\alpha+\beta)}{1+\tan^2(\alpha+\beta)} = \frac{1-(-6/7)^2}{1+(-6/7)^2} = \frac{1-36/49}{1+36/49} = \frac{13}{85}.