Note that I have assumed that the intended requirement was to show that:
#color(white)("XXX")(cos(x))/(1-sin(x)) - (cos(x))/(1+sin(x))=2tan(x)#
(as expressed in the question the expression is ambiguous)
#(cos(x))/(1-sin(x))-(cos(x))/(1+sin(x))#
#color(white)("XXX")=((cos(x)) * (1+sin(x)))/((1-sin(x)) * (1+sin(x)))-((cos(x)) * (1-sin(x)))/((1+sin(x)) * (1-sin(x)))#
#color(white)("XXX")=((cos(x)+cos(x) * sin(x))-(cos(x)-cos(x) * sin(x)))/(1-sin^2(x))#
#color(white)("XXX")=(2 * cos(x) * sin(x))/(cos^2(x))#
#color(white)("XXX")=2 * sin(x)/cos(x)#
#color(white)("XXX")=2 tan(x)#
