Question

If $\tan \theta +\sin \theta =m$ and $\tan \theta -\sin \theta =n$ then $4\sqrt{mn}=$

• A. $4\tan \theta \sin \theta$
• B. $4$
• C. $4\tan \theta$
• D. $4\sin \theta$

Answer 1

The correct option is A $4\tan \theta \sin \theta$

$mn=$$(\tan \theta +\sin \theta )(\tan \theta -\sin \theta )$
$={\tan }^2\theta -{\sin }^2\theta$
$=\frac{{\sin }^2\theta }{{\cos }^2\theta }-\frac{{\sin }^2\theta }{1}$
$=\frac{{\sin }^2\theta -{\sin }^2\theta {\cos }^2\theta }{{\cos }^2\theta }$
$=\frac{{\sin }^2\theta (1-{\cos }^2\theta )}{{\cos }^2\theta }=\frac{{\sin }^4\theta }{{\cos }^2\theta }$
$={\tan }^2\theta {\sin }^2\theta$
$\therefore 4\sqrt{mn}=4\tan \theta \sin \theta$