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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2022 · Mathematics · Trigonometric Ratios & Identities

If (1+ θ)(1+ 9 θ)=2 , then what is the value of (10 θ) ?

If (1+tan⁡θ)(1+tan⁡9θ)=2(1+\tan \theta)(1+\tan 9 \theta)=2, then what is the value of tan⁡(10θ)\tan (10 \theta) ?

Options

  1. A0
  2. B1
  3. C2
  4. DInfinite

Correct answer

B. 1

Step-by-step solution

∵(1+tan⁡θ)(1+tan⁡9θ)=2⇒1+tan⁡9θ+tan⁡θ+tan⁡θ⋅tan⁡9θ=2⇒tan⁡θ+tan⁡9θ=1−tan⁡θ⋅tan⁡9θ⇒tan⁡θ+tan⁡9θ1−tan⁡θ⋅tan⁡9θ=1⇒tan⁡(θ+9θ)=1⇒tan⁡10θ=1\begin{aligned} & \because(1+\tan \theta)(1+\tan 9 \theta)=2 \\ \Rightarrow & 1+\tan 9 \theta+\tan \theta+\tan \theta \cdot \tan 9 \theta=2 \\ \Rightarrow & \tan \theta+\tan 9 \theta=1-\tan \theta \cdot \tan 9 \theta \\ \Rightarrow & \frac{\tan \theta+\tan 9 \theta}{1-\tan \theta \cdot \tan 9 \theta}=1 \\ \Rightarrow & \tan (\theta+9 \theta)=1 \\ \Rightarrow & \tan 10 \theta=1\end{aligned}
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