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JEE Main 20198 AprilMorning Shift
Exam
JEE Main
Year
2019
Date
8 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

JEE Main 2019 · 8 April · Morning Shift · Mathematics · Trigonometric Ratios & Identities

If (α + β) = (3)/(5), ( α - β ) = (5)/(13) and 0 < α, β < (π)/(4),...

If cos⁡(α+β)=35,sin⁡(α−β)=513\cos \left(\alpha + \beta\right) = \dfrac{3}{5}, \sin ( \alpha - \beta ) = \dfrac{5}{13} and 0<α,β<π4,0 < \alpha, \beta < \dfrac{\pi}{4}, then tan⁡(2α)\tan \left(2\alpha\right) is equal to:

Options

  1. A2116\dfrac{21}{16}
  2. B6352\dfrac{63}{52}
  3. C3352\dfrac{33}{52}
  4. D6316\dfrac{63}{16}

Correct answer

D. 6316\dfrac{63}{16}

Step-by-step solution

cos⁡(α+β)=35⇒tan⁡(α+β)=43\cos \left(\alpha + \beta\right) = \dfrac{3}{5} ⇒ \tan \left(\alpha + \beta\right) = \dfrac{4}{3}

sin⁡(α−β)=513⇒tan⁡(α−β)=512\sin ( \alpha - \beta ) = \dfrac{5}{13} ⇒ \tan ( \alpha - \beta ) = \dfrac{5}{12}

Now,

tan⁡2α=tan⁡((α+β)+(α−β))\tan 2\alpha = \tan \left(\left(\alpha + \beta\right) + \left(\alpha - \beta\right)\right)

=tan⁡(α+β)+tan⁡(α−β)1−tan⁡(α+β)tan⁡(α−β)= \dfrac{\tan ( \alpha + \beta ) + \tan ( \alpha - \beta )}{1 - \tan ( \alpha + \beta ) \tan ( \alpha - \beta )}

=43+5121−43×512=6316= \dfrac{\dfrac{4}{3} + \dfrac{5}{12}}{1 - \dfrac{4}{3} \times \dfrac{5}{12}} = \dfrac{63}{16}

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