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NDA 2016 Maths · Mathematics · Trigonometric Functions and Identities · hard

The value of tan 20^°+tan 40^°+√3tan 20^°tan 40^° is

Single correct option

The value of tan⁡20∘+tan⁡40∘+3tan⁡20∘tan⁡40∘\tan20^\circ+\tan40^\circ+\sqrt3\tan20^\circ\tan40^\circ is
  1. A. 11
  2. B. 00
  3. C. 3\sqrt3
  4. D. 13\dfrac1{\sqrt3}
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Answer: (C) 3\sqrt3

Since 20∘+40∘=60∘20^\circ+40^\circ=60^\circ, tan⁡60∘=tan⁡20∘+tan⁡40∘1−tan⁡20∘tan⁡40∘=3\tan60^\circ=\dfrac{\tan20^\circ+\tan40^\circ}{1-\tan20^\circ\tan40^\circ}=\sqrt3. Cross-multiplying, tan⁡20∘+tan⁡40∘=3−3tan⁡20∘tan⁡40∘\tan20^\circ+\tan40^\circ=\sqrt3-\sqrt3\tan20^\circ\tan40^\circ. So the required expression equals 3\sqrt3.
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Original question written in the NDA 2016 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake