We know that, #tan2x=(2tanx)/(1-tan^2x)=(2t)/(1-t^2), where, t=tanx.#
#:. 2tanx-tan2x+2a=1-tan2xtan^2x,#
#rArr 2t-(2t)/(1-t^2)+2a=1-(2t)/(1-t^2)*t^2.#
Multiplying by #(1-t^2),# we get,
#2t(1-t^2)-2t+2a(1-t^2)=(1-t^2)-2t^3.#
#:. cancel(2t-2t^3-2t)+2a(1-t^2)=1-t^2cancel(-2t^3), or,#
# 2a(1-t^2)=(1-t^2).#
# :. 2a=(1-t^2)/(1-t^2)=1, if t^2ne1.#
#:. a=1/2, if t^2ne1.#
Now, #t^2=1 rArr t=tanx=+-1=tan(pmpi/4),#
# rArr x=npi+-pi/4, n in ZZ,# but, then, #tan2x# becomes undefined.
# :. t=tanxnepm1.#
#:. a=1/2.#