If =1 7 , =-1 3 , then 2 is equal to: - Vidyadip

MCQ

If $\tan\alpha=\frac{1}{7},\tan\beta=-\frac{1}{3},$ then $\cos2\alpha$ is equal to:

A
$\sin2\beta$
✓
$\sin4\beta$
C
$\sin3\beta$
D
$\cos^2\beta$

✓

Answer

Correct option: B.

$\sin4\beta$

it is given thet $\tan\alpha=\frac{1}{7}$ and $\tan\beta=\frac{1}{3}.$
Now,
$\tan\beta=\frac{2\tan\beta}{1-\tan^2\beta}$
$=\frac{2\times\frac{1}{3}}{1-\frac{1}{9}}$
$=\frac{\frac{2}{3}}{\frac{8}{9}}$
$=\frac{3}{4}$
$\therefore\tan(\alpha+2\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$
$=\frac{\frac{1}{7}+\frac{3}{4}}{1-\frac{1}{7}\times\frac{3}{4}}$
$=\frac{\frac{25}{28}}{\frac{25}{28}}$
$=1$
$\tan(\alpha+2\beta)=1=\tan\frac{\pi}{4}$
$\Rightarrow\alpha+2\beta=\frac{\pi}{4}$
$\Rightarrow\alpha=\frac{\pi}{4}-2\beta$
$\Rightarrow2\alpha=\frac{\pi}{2}-4\beta$
$\Rightarrow\cos2\alpha=\cos\big(\frac{\pi}{2}-4\beta\big)=\sin4\beta$
$\therefore\cos2\alpha=\sin4\beta$
Hence, the correct answer is option $B.$

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