Given equation
#sin3theta+cos3theta=1-sin2theta#
#=>3sintheta-4sin^3theta+4cos^3theta-3costheta-(1-sin2theta)=0#
#=>3(sintheta-costheta)-4(sin^3theta-cos^3theta)-(sin^2theta+cos^2theta-2sinthetacostheta)=0#
#=>3(sintheta-costheta)-4(sintheta-costheta)(sin^2theta+cos^2theta+sinthetacostheta)-(sin^2theta+cos^2theta-2sinthetacostheta)=0#
#=>3(sintheta-costheta)-4(sintheta-costheta)(1+sinthetacostheta)-(sintheta-costheta)^2=0#
#=>(sintheta-costheta)(3-4-4sinthetacostheta-sintheta+costheta)=0#
#=>(sintheta-costheta)(costheta-1-4sinthetacostheta-sintheta)=0#
So
#sintheta-costheta=0#
#=>sintheta=costheta#
#=>tantheta=1->"option A possible"#
Checking opotion B, #costheta=0# or #theta =90# for 2nd equation
#costheta-1-4sinthetacostheta-sintheta#
#=cos90-1-4*sin90*cos90-sin90#
#=0-1-0-1=-2#
So option B is not possible
Checking opotion C #tan(theta/2)=-1# or #theta/2 =-45# or #theta=-90# for 2nd equation
#costheta-1-4sinthetacostheta-sintheta#
#=cos(-90)-1-4*sin(-90)*cos(-90)-sin(-90)#
#=0-1-0+1=0#
So option C is possible
Checking opotion D, #cos(theta/2)=0# or #theta =180^@# for 2nd equation
#costheta-1-4sinthetacostheta-sintheta#
#=cos180-1-4*sin180*cos180-sin180#
#=-1-1-0-0=-2#
So option D is not possible