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Question: \(2\tan^{- 1}\left( \cos ec\tan^{- 1}\sqrt{3} - {tancot}^{- 1}\sqrt{3} \right)\) is equal to...

2tan⁡−1(cos⁡ectan⁡−13−tancot−13)2\tan^{- 1}\left( \cos ec\tan^{- 1}\sqrt{3} - {tancot}^{- 1}\sqrt{3} \right) is equal to

A

π/16

B

π/6

C

π/3

D

π/2

Answer

π/3

Explanation

Solution

The given expression is equal to

2tan⁡−1(cos⁡ecπ3−tan⁡π6)2\tan^{- 1}\left( \cos ec\frac{\pi}{3} - \tan\frac{\pi}{6} \right)

= 2tan⁡−1(23−13)=2tan⁡−113=2 x π6=π32\tan^{- 1}\left( \frac{2}{\sqrt{3}} - \frac{1}{\sqrt{3}} \right) = 2\tan^{- 1}\frac{1}{\sqrt{3}} = 2\text{ x }\frac{\pi}{6} = \frac{\pi}{3}