Question

If $\tan \theta+\sin \theta=m$ and $\tan \theta-\sin \theta=n$, then

• A. $m^{2}-n^{2}=4 m n$
• B. $m^{2}+n^{2}=4 m n$
• C. $m^{2}-n^{2}=m^{2}+n^{2}$
• D. $m^{2}-n^{2}=4 \sqrt{m n}$

Answer 1

Answer: (D)

From the given relations,
$m+ n=2 \tan \theta, m-n=2 \sin \theta$.
Thus, $m^{2}-n^{2}=4 \tan \theta \sin \theta$ ...(i)
Also $\sqrt{m n}=4 \sqrt{\tan ^{2} \theta-\sin ^{2} \theta}$ ...(ii)
$=4 \sin \theta \tan \theta$
From Eqs. (i) and (ii), we get $m^{2}-n^{2}=4 \sqrt{m n}$.