Question 1025818
==> {{{tanx = (m+n)/2}}} and {{{sinx = (m-n)/2}}}

==> {{{tanx*sinx = (m^2 - n^2)/4}}} ==> {{{4tanx*sinx = (m^2 - n^2) = (4sin^2(x))/cos(x)}}} 

= {{{4sqrt(sin^4(x)/cos^2(x)) = 4sqrt((sin^2(x)*sin^2(x))/cos^2(x))}}}

= {{{4sqrt((sin^2(x)*(1-cos^2(x)))/cos^2(x))}}}

= {{{4sqrt(sin^2(x)/cos^2(x)-sin^2(x)) = 4sqrt(tan^2(x)-sin^2(x)) }}}

= {{{4sqrt((tanx+sinx)(tanx-sinx)) = 4sqrt(mn)}}}

Therefore {{{m^2 - n^2=4sqrt(mn)}}}