$\begin{array}{lll}\text { We have } & \tan \theta+\sin \theta=m \\ \text { and } & \tan \theta-\sin \theta=n\end{array}$ (i) and $\tan \theta-\sin \theta=n$ Now, $m+n=\tan \theta+\sin \theta+\tan \theta-\sin \theta$ $\Rightarrow m+n=2 \tan \theta$
Also, $m-n=\tan \theta+\sin \theta-\tan \theta+\sin \theta$ $\Rightarrow m-n=2 \sin \theta$
Multiplying Eqns. (iii) and (iv), we get:
$$
\begin{aligned}
&(m+n)(m-n)=4 \sin \theta \cdot \tan \theta \\
&\Rightarrow m^2-n^2=4 \sin \theta \cdot \tan \theta
\end{aligned}
$$
Hence proved.