Multiple choice

If $\tan{(\displaystyle\frac{\alpha}{2})}$ and $\tan{(\displaystyle\frac{\beta}{2})}$ are the roots of the equation $8{x}^{2}-26x+15=0$ then $\cos{(\alpha +\beta)}$ is equal to-

  1. $-\displaystyle\frac{627}{725}$
  2. $\displaystyle\frac{627}{725}$
  3. $-\displaystyle\frac{725}{625}$
  4. $-1$
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A Correct answer
Explanation

Let tan(a/2) = u, tan(b/2) = v. u+v = 26/8 = 13/4. uv = 15/8. cos(a+b) = (1 - tan^2((a+b)/2)) / (1 + tan^2((a+b)/2)). tan((a+b)/2) = (u+v)/(1-uv) = (13/4) / (1 - 15/8) = (13/4) / (-7/8) = -26/7. tan^2((a+b)/2) = 676/49. cos(a+b) = (1 - 676/49) / (1 + 676/49) = (-627/49) / (725/49) = -627/725.

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