Given #tan theta + sin theta=m#, #tan theta-sin theta=n#
Now
#m^2-n^2#
#=(tantheta+sintheta)^2-(tantheta-sintheta)^2#
#=4tanthetasintheta#
#=4sqrt(tan^2thetasin^2theta)#
#=4sqrt(tan^2theta(1-cos^2theta))#
#=4sqrt(tan^2theta-tan^2thetacos^2theta)#
#=4sqrt(tan^2theta-sin^2theta/cos^2thetacos^2theta)#
#=4sqrt(tan^2theta-sin^2theta)#
#=4sqrt((tantheta+sintheta)(tantheta-sin theta))#
#=4sqrt(mn)#