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JEE Main 2019 · 8 April · Shift I

JEE Main 2019 Mathematics question on Trigonometry

If cos⁡(α+β)=35\cos(\alpha + \beta) = \dfrac{3}{5}, sin⁡(α−β)=513\sin(\alpha - \beta) = \dfrac{5}{13} and 0<α,β<π40 < \alpha, \beta < \dfrac{\pi}{4}, then tan⁡(2α)\tan(2\alpha) is equal to
  1. A.
    6352\dfrac{63}{52}
  2. B.
    6316\dfrac{63}{16}
  3. C.
    2116\dfrac{21}{16}
  4. D.
    3352\dfrac{33}{52}
Show answer and solution

Correct option: B

Solution

Since cos(α+β) = 3/5 with (α+β) in (0, π/2), we get sin(α+β) = 4/5, so tan(α+β) = 4/3.

Since sin(α-β) = 5/13 with (α-β) in (-π/4, π/4), we get cos(α-β) = 12/13, so tan(α-β) = 5/12.

Now 2α = (α+β) + (α-β), so:

tan(2α) = [tan(α+β) + tan(α-β)] / [1 - tan(α+β)·tan(α-β)]
= [(4/3) + (5/12)] / [1 - (4/3)(5/12)]
= [(16/12) + (5/12)] / [1 - 20/36]
= (21/12) / (16/36)
= (21/12) × (36/16)
= 756/192
= 63/16.

[JEE Main 2019 · 8 April · Shift I]

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