Use the identity, cos A = 1/sec A and substitute the value of cos A in the left hand side of the equation:
(1 + cos A)/(1-cos A) = (1+1/sec A)/(1-1/sec A) = [(secA +1)/sec A]/[(secA - 1)/sec A]
cancelling out the sec A from both numerator and denominator, we get
(sec A +1)/(sec A -1) = right hand side. Hence proved.