Proof Question

$\mathrm{Use} \mathrm{the} \mathrm{result} 1+\mathrm{cos2\theta }=2{\cos }^2\theta \mathrm{repeatedly}.$$\mathrm{Hence} 2+2\mathrm{cos8\theta }=2(1+\mathrm{cos8\theta })=2\left(2{\cos }^{2}4\theta \right)=4{\cos }^{2}4\theta$$\sqrt{2+2\mathrm{cos8\theta }}=\sqrt{4{\cos }^{2}4\theta }=2\mathrm{cos4\theta }$$$$\mathrm{Again} 2+2\mathrm{cos4\theta }=2(1+\mathrm{cos4\theta })=2\left(2{\cos }^{2}2\theta \right)=4{\cos }^{2}2\theta$$\sqrt{2+2\mathrm{cos4\theta }}=\sqrt{4{\cos }^{2}2\theta }=2\mathrm{cos2\theta }$$$$\mathrm{Now} \mathrm{apply} \mathrm{again} \mathrm{the} \mathrm{same} \mathrm{result}.$">