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NDA 2022 Maths · Mathematics · Trigonometric Functions and Identities · hard

The value of tan 20^° tan 40^° tan 80^° is

Single correct option

The value of tan⁡20∘ tan⁡40∘ tan⁡80∘\tan20^\circ\,\tan40^\circ\,\tan80^\circ is
  1. A. 11
  2. B. 13\dfrac{1}{\sqrt3}
  3. C. 3\sqrt3
  4. D. 33
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Answer: (C) 3\sqrt3

Use tan⁡θ tan⁡(60∘−θ) tan⁡(60∘+θ)=tan⁡3θ\tan\theta\,\tan(60^\circ-\theta)\,\tan(60^\circ+\theta)=\tan3\theta. With θ=20∘\theta=20^\circ: tan⁡20∘tan⁡40∘tan⁡80∘=tan⁡60∘=3\tan20^\circ\tan40^\circ\tan80^\circ=\tan60^\circ=\sqrt3. A common slip is to treat it like the sine product sin⁡20∘sin⁡40∘sin⁡80∘=38\sin20^\circ\sin40^\circ\sin80^\circ=\dfrac{\sqrt3}{8} and pick a fraction; the tangent identity has no 14\tfrac14 factor.
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Original question written in the NDA 2022 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