Question

If $\tan \theta +\sin \theta =m$ and $\tan \theta -\sin \theta =n$ then prove that ${\tan }^2\theta -{\sin }^2\theta =mn$.

Answer 1

Given, $\tan \theta +\sin \theta =m$......(1) and $\tan \theta -\sin \theta =n$......(2).

Now, multiplying (1) and (2) we get,

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

or, $\frac{{\sin }^2\theta }{{\cos }^2\theta }-{\sin }^2\theta =mn$

or, ${\sin }^2\theta \frac{1-{\cos }^2\theta }{{\cos }^2\theta }=mn$

or, ${\sin }^2\theta .\frac{{\sin }^2\theta }{{\cos }^2\theta }=mn$ [ Since ${\sin }^2\theta =1-{\cos }^2\theta$]

or, ${\tan }^2\theta .{\sin }^2\theta =mn$.