Multiple choice

If tan⁡ α/2 and tan⁡ β/2 are the roots of the equation 8x2- 26x+15=0, then cos⁡ (α+β) is equal to

    • 627/725

  2. 627/725
    • 1

  4. none of these
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A Correct answer
Explanation

Let u = tan(alpha/2) and v = tan(beta/2). u+v = 26/8 = 13/4, uv = 15/8. tan((alpha+beta)/2) = (u+v)/(1-uv) = (13/4) / (1 - 15/8) = (13/4) / (-7/8) = -26/7. Then cos(alpha+beta) = (1 - tan^2(theta)) / (1 + tan^2(theta)) where theta = (alpha+beta)/2. Calculation: (1 - 676/49) / (1 + 676/49) = -627/725.

AI explanation

By Vieta's formulas, the sum and product of the roots tan(alpha/2) and tan(beta/2) are 26/8 = 13/4 and 15/8, respectively. Using the tangent addition formula, tan((alpha+beta)/2) equals (13/4) / (1 - 15/8) = -26/7. Applying the double angle formula for cosine gives cos(alpha+beta) = (1 - tan^2((alpha+beta)/2)) / (1 + tan^2((alpha+beta)/2)) = (1 - 676/49) / (1 + 676/49). This simplifies to (-627/49) / (725/49) = -627/725.

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