#cosx+sinx=cos2x+sin2x#.
#:. cosx-cos2x=sin2x-sinx#.
#:. -2sin((x+2x)/2)sin((x-2x)/2)=2cos((2x+x)/2)sin((2x-x)/2)#.
#:. +sin(3/2x)sin(+1/2x)=cos(3/2x)sin(1/2x)#.
#:. sin(3/2x)sin(1/2x)-cos(3/2x)sin(1/2x)=0#.
#:. sin(1/2x)[sin(3/2x)-cos(3/2x)]=0#.
#:. sin(1/2x)=0, or, sin(3/2x)=cos(3/2x)#.
Case 1 : #sin(1/2x)=0#.
#sin(1/2x)=0 rArr 1/2x=kpi rArr x=2kpi, k in ZZ.#
Case 2 : #sin(3/2x)=cos(3/2x)#.
Note that #cos(3/2x)" can not be "0,# because, in that case, by the
virtue of the eqn., #sin(3/2x)" will also be "0,# contradicting,
#sin^2(3/2x)+cos^2(3/2x)=1#.
So, dividing by #cos(3/2x)ne0#, we get,
#tan(3/2x)=1=tan(pi/4)," giving, "#
#3/2x=kpi+pi/4 rArr x=2/3kpi+pi/6=(4k+1)pi/6, k in ZZ.#
Altogether, The Soln. Set =#{2kpi}uu{(4k+1)pi/6}, k in ZZ#.
