Question

If $\tan \theta +\sin \theta =m$ and $\tan \theta -\sin \theta =n$, then the value of $m^2-n^2$ is?

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

Answer 1

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

$\tan \theta +\sin \theta =m$

$\tan \theta -\sin \theta =n$

$m^2-n^2=4\tan \theta \sin \theta$

$mn={\tan }^2\theta -{\sin }^2\theta$

$⟹\frac{{\sin }^2\theta -{\sin }^2\theta {\cos }^2\theta }{{\cos }^2\theta }$

$⟹\frac{{\sin }^2\theta -{\sin }^2\theta (1-{\sin }^2\theta )}{{\cos }^2\theta }$

$⟹\frac{{\sin }^4\theta }{{\cos }^2\theta }$

$⟹{\tan }^2\theta \times {\sin }^2\theta$

$m^2-n^2=4\sqrt{mn}$