Question

If $\tan \alpha =\frac{1}{7}$ and $\sin \beta =\frac{1}{\sqrt{10}}$ where $0<\alpha ,\beta <\frac{\pi }{2}$, then $2\beta$ is equal to

• A. $\frac{\pi }{4}-\alpha$
• B. $3\pi -\alpha$
• C. $\frac{\pi }{8}-\alpha$
• D. $\frac{3\pi }{8}-\frac{\alpha }{2}$

Answer 1

Answer: (A)

$\frac{\pi }{4}-\alpha$

Given $\tan \alpha =\frac{1}{7}$
and $\sin \beta =\frac{1}{\sqrt{10}}$
$\tan \beta =\frac{1}{3}$
Now, $\tan 2\beta =\frac{2\tan \beta }{1-{\tan }^2\beta }$
$\Rightarrow \tan 2\beta =\frac{3}{4}$
Now, $\tan (\alpha +2\beta )=\frac{\tan \alpha +\tan 2\beta }{1-\tan \alpha \tan 2\beta }$
$\Rightarrow \tan (\alpha +2\beta )=\frac{\frac{1}{7}+\frac{3}{4}}{1-\frac{1}{7}\times \frac{3}{4}}$
$\Rightarrow \tan (\alpha +2\beta )=1$
$\Rightarrow \alpha +2\beta =\frac{\pi }{4}$
$\Rightarrow 2\beta =\frac{\pi }{4}-\alpha$