#(1-cos(theta))(1+sec(theta))=sin^2(theta)#
Using FOIL, multiply out the left hand side.
#1+sec(theta)-cos(theta)-cos(theta)sec(theta)#
Convert the #sec#
#1 +1/cos(theta)-cos(theta)-cos(theta) 1/cos(theta)#
#cancel(1 )+ 1/cos(theta) - cos(theta) - cancel(1)#
#1/cos(theta) - cos^2(theta)/cos(theta)#
#(1-cos^2(theta))/cos(theta)#
#sin^2(theta)/cos(theta)# or #sin(theta)tan(theta)#
This is not equivalent to the right hand side, #sin^2(theta)#. You can also see that they are not equivalent by looking at the graphs of both sides.
