If θ+ (θ+(π)/(3))+ (θ+(2 π)/(3))=3 , then which of the following is equal to 1 ?

Correct answer: B. 3 θ Given, aligned & θ+ (θ+(π)/(3))+ (θ+(2 π)/(3))=3 & θ+ θ+ 3 1- 3 θ + θ- 3 1+ 3 θ =3 & θ+(8 θ)/(1-3 ^2 θ)=3 & (9 θ-3 ^3 θ)/(1-3 ^2 θ)=3 & 3 3 θ=3 3 θ=1 & aligned Hence, option (2) is correct.

JEE Main

Exam: JEE Main
Subject: Mathematics
Chapter: Trigonometric Ratios & Identities

JEE Main · Mathematics · Trigonometric Ratios & Identities

If θ+ (θ+(π)/(3))+ (θ+(2 π)/(3))=3 , then which of the following is...

\tan \theta+\tan \left(\theta+\frac{\pi}{3}\right)+\tan \left(\theta+\frac{2 \pi}{3}\right)=3

, then which of the following is equal to 1 ?

Options

A $\tan 2 \theta$
B $\tan 3 \theta$
C $\tan ^2 \theta$
D $\tan ^3 \theta$

Correct answer

B. $\tan 3 \theta$

Step-by-step solution

Given,

\begin{aligned}& \tan \theta+\tan \left(\theta+\frac{\pi}{3}\right)+\tan \left(\theta+\frac{2 \pi}{3}\right)=3 \\ & \Rightarrow \tan \theta+\frac{\tan \theta+\sqrt{3}}{1-\sqrt{3} \tan \theta}+\frac{\tan \theta-\sqrt{3}}{1+\sqrt{3} \tan \theta}=3 \\ & \Rightarrow \quad \tan \theta+\frac{8 \tan \theta}{1-3 \tan ^2 \theta}=3 \\ & \Rightarrow \quad \frac{9 \tan \theta-3 \tan ^3 \theta}{1-3 \tan ^2 \theta}=3 \\ & \Rightarrow \quad 3 \tan 3 \theta=3 \Rightarrow \tan 3 \theta=1 \\ & \end{aligned}

Hence, option (2) is correct.

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