Rewrite #tanx# as #sinx/cosx#:
#(sinx + cosx)/(sinx/cosx) + (1 - sinx)/sinx = cosx#
#(cosx(sinx + cosx))/sinx + (1 - sinx)/sinx = cosx#
#(cosxsinx + cos^2x + 1 - sinx)/sinx = cosx#
#cosxsinx + cos^2x + 1 - sinx = sinxcosx#
#cos^2x + 1 - sinx = sinxcosx- sinxcosx#
#cos^2x+ 1 - sinx = 0#
Use #sin^2x + cos^2x = 1#:
#1 - sin^2x + 1 - sinx = 0#
#0 =sin^2x + sinx - 2#
Factor:
#0 = (sinx + 2)(sinx - 1)#
#sinx = -2 and 1#
There is no solution to #sinx = -2#. However, #sinx = 1# is solvable.
#x = pi/2 #
However, this is extraneous, since #tan(pi/2)# is not defined in the real number system, aka:
#tan(pi/2) = sin(pi/2)/cos(pi/2) = 1/0 = "undefined"#
Therefore, this equation has no solution. This can be symbolized with an empty solution set #{}#.
Hopefully this helps!
