Question

If $\tan \theta =a\ne 0,\tan 2\theta =b\ne 0$ and $\tan \theta +\tan 2\theta =\tan 3\theta$ then

• A. $a=b$
• B. $ab=1$
• C. $a+b=0$
• D. $b=2a$

Answer 1

The correct option is C $a+b=0$

$a=\tan \theta \&b=\tan 2\theta$ where $a\ne 0,b\ne 0$ ...(1)
$\tan \theta +\tan 2\theta =\tan 3\theta$ ...(2)
$A-B=C\tan (A-B)=\tan C$
$\Rightarrow \frac{\tan A-\tan B}{1+\tan A\tan B}=\tan C$
$\Rightarrow \tan A-\tan B-\tan C=\tan A\tan B\tan C$
$\Rightarrow \tan 3\theta -\tan 2\theta -\tan \theta =\tan 3\theta \tan 2\theta \tan \theta$
$\Rightarrow ab\tan 3\theta =0$ ...{ from (2)}
$\because a\ne 0,b\ne 0\therefore \tan 3\theta =0\Rightarrow \tan \theta +\tan 2\theta =0$
$\Rightarrow a+b=0$ ...{ from (1)}

Ans: C