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NDA 2014
Exam
NDA
Year
2014
Subject
Mathematics
Chapter
Trigonometric Equations

NDA 2014 · Mathematics · Trigonometric Equations

Which one of the following is one of the solutions of the equation...

Which one of the following is one of the solutions of the equation tan⁡2θ⋅tan⁡θ=1\tan 2 \theta \cdot \tan \theta=1 ?

Options

  1. Aπ12\frac{\pi}{12}
  2. Bπ6\frac{\pi}{6}
  3. Cπ4\frac{\pi}{4}
  4. Dπ3\frac{\pi}{3}

Correct answer

B. π6\frac{\pi}{6}

Step-by-step solution

tan⁡2θ⋅tan⁡θ=1⇒2tan⁡θ1−tan⁡2θ×tan⁡θ=12tan⁡2θ=1−tan⁡2θ3tan⁡2θ=1⇒tan⁡2θ=13⇒tan⁡θ=13=tan⁡30∘⇒θ=30∘=π6\begin{aligned} & \tan 2 \theta \cdot \tan \theta=1 \\ & \Rightarrow \frac{2 \tan \theta}{1-\tan ^2 \theta} \times \tan \theta=1 \\ & 2 \tan ^2 \theta=1-\tan ^2 \theta \\ & 3 \tan ^2 \theta=1 \\ & \Rightarrow \quad \tan ^2 \theta=\frac{1}{3} \\ & \Rightarrow \quad \tan \theta=\frac{1}{\sqrt{3}}=\tan 30^{{}^{\circ}} \\ & \Rightarrow \quad \theta=30^{{}^{\circ}}=\frac{\pi}{6} \end{aligned}
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