#tan (A-B) /tanA +sin^2C /sin^2A=1 #
#=>sin^2C /sin^2A=1-tan (A-B) /tanA #
#=>sin^2C /sin^2A=1-(sin(A-B) cosA)/(cos(A-B)sinA ) #
#=>sin^2C /sin^2A=(sinAcos(A-B)-sin(A-B) cosA)/(cos(A-B)sinA ) #
#=>sin^2C=sin(A-A+B)/(cos(A-B)sinA ) xxsin^2A#
#=>sin^2C=(sinAsinB)/cos(A-B)#
#=>csc^2C=cos(A-B)/(sinAsinB)#
#=>csc^2C=(cosAcosB+sinAsinB)/(sinAsinB)#
#=>csc^2C=(cosAcosB)/(sinAsinB)+(sinAsinB)/(sinAsinB)#
#=>1+cot^2C=cotAcotB+1#
#=>cotAcotB=cot^2C#
#=>tanAtanB=tan^2C#
Proved