Substituting
#cos 2x=cos^2 x-sin^2 x=2cos^2 x -1#
And
#sin 2x=2sin x*cos x#
The expression becomes
#rarr1-2cos^2 x+1+(sin x/cos x)/(1-sinx/cos x)=1+2sinx*cosx#
#rarr2-2cos^2 x+(sinx/cancel(cosx))(cancel(cosx)/(cosx-sin x))-1-
2sinx*cosx=0#
#rarr2sin^2 x(cosx-sinx)+sinx-(1+2sinx*cosx)(cosx-sinx)=0#
#rarr2sin^2 x*cosx-2sin^3 x+sin x -cosx+sinx-2sinx*cos^2 x+2sin^2
x*cosx=0#
#rarr4sin^2 x*cosx-2sin^3 x+2sinx-cosx-2sinx*cos^2x=0#
#rarr4sin^2 x*cosx-2sin^3 x+2sinx-cosx-2sinx*(1-sin^2 x)=0#
#rarr4sin^2 x*cosx-cancel(2sin^3 x)+cancel(2sinx)-cosx-
cancel(2sinx)+cancel(2sin^3 x)=0#
#rarr4sin^2 x*cosx-cosx=0#
#rarrcosx(4sin^2x-1)=0#
#cos x=0# => #x=90^@ +k*180^@ -># but in the original expression there's #tgx# and #tgx# is undefined for #x=90^@ or x=270^@# so we must outrightly reject this solution or only admit it when #x->90^@#
#4sin^2 x-1=0# #=># #sin^2 x=1/4# #=># #sin x=1/2# #=># #x=30^@+k*360^@# or #x=150^@+k*360^@#, where #k in NN#
