#color(blue)(cosx+cos^2x=1,#
then #color(green)(cosx=1-cos^2x=sin^2x.#
#sin^12x+3sin^10x+3 sin^8x+sin^6x+2 sin^4x+2sin^2x-2 #
#= cos^6x+3cos^5x+3 cos^4x+cos^3x+2 cos^2x+2cosx-2 #
#=cos^6x+3cos^5x+3 cos^4x+cos^3x #
#=cos^4x (cos^2x+3) + cos^3x(3cos^2+1) #
#= cos^4x (1-cosx+3) +cos^3x(3-3cosx+1) #
#= (1-cosx)^2 (4-cosx)+cosx((1-cosx)(4-3cosx) #
#= (1-2cosx+1-cosx)(4-cosx)+cosx(4-7cosx+3-cos x #
#= (2-3cosx)(4-cosx)+ cosx(7-10cosx) #
#= 8-14cosx+3cos^2x+7cosx-10cos^2x #
#= 8-7cosx-7cos^2x=8-7(1) #
#= 1#