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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NTA Abhyas 2020 · Mathematics · Trigonometric Ratios & Identities

If 0 < α < (π)/(16) and ( 1 + tan α )(1 + 4α) = 2 , then the value...

If 0<α<π160 < \alpha < \dfrac{\pi}{16} and (1+tanα)(1+tan⁡4α)=2( 1 + tan \alpha )\left(1 + \tan 4\alpha\right) = 2, then the value of α\alpha is equal to

Options

  1. Aπ18\dfrac{\pi}{18}
  2. Bπ20\dfrac{\pi}{20}
  3. Cπ24\dfrac{\pi}{24}
  4. Dπ30\dfrac{\pi}{30}

Correct answer

B. π20\dfrac{\pi}{20}

Step-by-step solution

We have
1+tanα+tan⁡4α+tanαtan4α=21 + tan \alpha + \tan 4\alpha + tan \alpha tan 4\alpha = 2
tanα+tan⁡4α=1−tanαtan4αtan \alpha + \tan 4\alpha = 1 - tan \alpha tan 4\alpha
tanα+tan⁡4α1−tanαtan4α=1\dfrac{tan \alpha + \tan 4\alpha}{1 - tan \alpha tan 4\alpha} = 1
tan⁡(α+4α)=1\tan \left(\alpha + 4\alpha\right) = 1
5α=π4(since,0<α<π16)5\alpha = \dfrac{\pi}{4} \left(since, 0 < \alpha < \dfrac{\pi}{16}\right)
Hence, α=π20\alpha = \dfrac{\pi}{20}
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