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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2024 · Mathematics · Trigonometric Ratios & Identities

If α and β are the roots of the equation x^2-6 x+8=0 , then what is...

If tan⁡α\tan \alpha and tan⁡β\tan \beta are the roots of the equation x2−6x+8=0x^2-6 x+8=0, then what is the value of cos⁡(2α+2β)?\cos (2 \alpha+2 \beta) ?

Options

  1. A1375\frac{13}{75}
  2. B1385\frac{13}{85}
  3. C1785\frac{17}{85}
  4. D1985\frac{19}{85}

Correct answer

B. 1385\frac{13}{85}

Step-by-step solution

∵tan⁡α+tan⁡β=6\because \tan \alpha+\tan \beta=6tan⁡α⋅tan⁡β=8tan⁡(α+β)=tan⁡α+tan⁡β1−tan⁡α⋅tan⁡β=61−8=6−7cos⁡(2α+2β)=cos⁡2(α+β)=1−tan⁡2(α+β)1+tan⁡2(α+β)=1−36491+3649=1385\begin{aligned} \tan \alpha \cdot \tan \beta & =8 \\ \tan (\alpha+\beta) & =\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \cdot \tan \beta} \\ & =\frac{6}{1-8}=\frac{6}{-7} \\ \cos (2 \alpha+2 \beta) & =\cos 2(\alpha+\beta) \\ & =\frac{1-\tan ^2(\alpha+\beta)}{1+\tan ^2(\alpha+\beta)} \\ & =\frac{1-\frac{36}{49}}{1+\frac{36}{49}} \\ & =\frac{13}{85}\end{aligned}$$
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