Question

If tan $ 3\theta+tan5\theta+tan 3\theta=1 $ , then $ \theta $ is equal to

• A. $ \frac{n\pi}{8}+\frac{\pi}{16}, n \in Z $
• B. $ \frac{n\pi}{4}+\frac{\pi}{32}, n \in Z $
• C. $ \frac{n\pi}{8}+\frac{\pi}{32}, n \in Z $
• D. $ \frac{n\pi}{8}+\frac{\pi}{12}, n \in Z $

Answer 1

Answer: (C)

We have, $tan\,30 + tan\,50 + tan\,30\,tan\,50 = 1$
$\Rightarrow tan\,30 + tan\,50 = 1 - tan\,30\,tan\,50$
$\Rightarrow \frac{tan\,3\theta + tan\,5\theta}{1-tan\,3\theta\,tan\,5\theta} = 1$
$\Rightarrow tan (3\theta + 5\theta) = 1$
$\Rightarrow tan\,8\theta = \tan \frac{\pi}{4}$
$\Rightarrow 2\theta = n \pi + \frac{\pi}{4}, n \in Z$
$\Rightarrow \theta = \frac{n\pi}{8} + \frac{\pi}{32} , n\in Z$