Question

Assertion($A$): $a=\tan \theta ,b=\tan 2\theta ,a\ne 0,b\ne 0$ and $\tan \theta +\tan 2\theta =\tan 3\theta$, then $a+b=0$
Reason $\left(R\right)$ : lf $A-B=C$, then $\tan A-\tan B-\tan C=\tan A\tan B\tan C$

• A. A is true, R is true and R is correct explanation of A
• B. A is true, R is true and R is not correct explanation of A
• C. A is true, R is false
• D. A is false, R is true.

Answer 1

Answer: (A)

A is true, R is true and R is correct explanation of A

$a=\tan \theta \&b=\tan 2\theta$ 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)}
Hence, option 'A' is correct.