If tan theta + sin theta=m, …

L.H.S. ${m^2} - {n^2}$ = ${\left( {\tan \theta + \sin \theta } \right)^2} - {\left( {\tan \theta - \sin \theta } \right)^2}$

= $\left( {\tan \theta + \sin \theta + \tan \theta - \sin \theta } \right)\left( {\tan \theta + \sin \theta - \tan \theta + \sin \theta } \right)$  [Since ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$]

= $\left( {2\tan \theta } \right)\left( {2\sin \theta } \right)$

= $4 \times {{\sin \theta } \over {\cos \theta }} \times \sin \theta$

= ${{4{{\sin }^2}\theta } \over {\cos \theta }}$

R.H.S. $4\sqrt {mn}$ = $4\sqrt {\left( {\tan \theta + \sin \theta } \right)\left( {\tan \theta - \sin \theta } \right)}$

= $4\sqrt {{{\tan }^2}\theta - {{\sin }^2}\theta }$                                                                          [Since ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$]

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

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

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

= $4\sqrt {{{{{\sin }^2}\theta .{{\sin }^2}\theta } \over {{{\cos }^2}\theta }}}$

= ${{4{{\sin }^2}\theta } \over {\cos \theta }}$

L.H.S. = R.H.S.