Question

If $\tan \theta +\sin \theta =m$ and $\tan \theta -\sin \theta =n$ then $m^2-n^2$ is equal to

• A. 4mn
• B. $4\sqrt{mn}$
• C. 2mn
• D. $2\sqrt{mn}$

Answer 1

The correct option is B $4\sqrt{mn}$

$m^2-n^2={(\tan \theta +\sin \theta )}^2-{(\tan \theta -\sin \theta )}^2=4\tan \theta \sec \theta$
Also
$\sqrt{mn}=\sqrt{{\tan }^2\theta -{\sin }^2\theta }$
$=\sqrt{{\sin }^2\theta ({\sin }^2\theta -1)}$
$=\sin \theta \tan \theta$
$\Rightarrow m^2-n^2=4\sqrt{nm}$