Question

if cosec A+ secA =cosecB +sec B prove that tanA tan B =cot(A+B /2)

Answer 1

here
cosec A+ secA =cosecB +sec B
or,1/sinA+1/cosA=1/sinB+1/cosB
or,1/sinA-1/sinB=1/cosB-1/cosA
or.(sinB-sinA)/sinAsinB=(cosA-cosB)/cosAcosB
or,sinAsinB/cosAcosB=(sinB-sinA)/(cosA-cosB)
or,tanAtanB=[2cos(A+B)sin(B-A)]/[-2sin{(A+B)/2]*[sin(A-B)/2}]
or,tanAtanB=[2cos(A+B)sin(B-A)]/[-2sin{(A+B)/2][-sin(B-A)/2]/
or,tanAtanB=cos[(A+B)/2]/[sin(A+B)/2]
          tanAtanB=cos(A+B)/2 
                                             PROVED