If tan theta + sin theta=m, …

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

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

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

= {tex}4 \times {{\sin \theta } \over {\cos \theta }} \times \sin \theta {/tex}

= {tex}{{4{{\sin }^2}\theta } \over {\cos \theta }}{/tex}

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

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

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

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

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

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

= {tex}{{4{{\sin }^2}\theta } \over {\cos \theta }}{/tex}

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