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.

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