tanA+sinA =m tanA -sinA =n then …

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

L.H.S = $m^{2}-n^{2}$

$=(tan\theta+sin\theta)^{2}-(tan\theta-sin\theta)^{2}$

$=tan^{2}\theta+sin^{2}\theta+2tan\theta sin\theta-[tan^{2}\theta+sin^{2}\theta-2tan\theta sin\theta]$

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

$=4tan\theta sin\theta$

R.H.S = 4$\sqrt{mn}$

$=4\sqrt{(tan\theta+sin\theta)(tan\theta-sin\theta)}$

$=4\sqrt{tan^{2}\theta-sin^{2}\theta}$

$=4\sqrt{\frac{sin^{2}\theta}{cos^{2}\theta}-sin^{2}\theta}$

$=4\sqrt{\frac{sin^{2}\theta-sin^{2}\theta cos^{2}\theta}{cos^{2}\theta}}$

$=\frac{4}{cos\theta}\sqrt{sin^{2}\theta-sin^{2}\theta cos^{2}\theta}$

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

$=\frac{4}{cos\theta} \times sin\theta \times \sqrt{sin^{2}\theta}$

$=4tan\theta sin\theta$

Hence, L.H.S = R.H.S