How do you solve ##Cos 2 theta = cos theta##?

##theta=(2n+1)pi/2## or ##theta=2npi+-(2pi)/3##, where ##n## is an integer.

##cos2theta=costheta##

or ##2cos^2theta-1=costheta## (using formula for ##cos2theta##)

the above becomes ##2cos^2theta-costheta-1=0##

Now using quadratic formula

##costheta=(-(-1)+-sqrt((-1)^2-4*2*(-1)))/(2*2)##

or ##costheta=(1+-sqrt(1+8))/4=(1+-3)/4##

Hence ##costheta=1=cos0## or ##costheta=-1/2=cos((2pi)/3)##

Hence ##theta=(2n+1)pi/2## or ##theta=2npi+-(2pi)/3##, where ##n## is an integer.