1. The problem is to simplify the expression $$\frac{\cos \theta}{2} \pm \frac{5 \sin \theta}{5} \cos 60^\circ$$. 2. Recall that $$\cos 60^\circ = \frac{1}{2}$$. 3. Substitute $$\cos 60^\circ$$ into the expression: $$\frac{\cos \theta}{2} \pm \frac{5 \sin \theta}{5} \times \frac{1}{2}$$. 4. Simplify the fraction $$\frac{5 \sin \theta}{5}$$ by canceling 5: $$\frac{\cos \theta}{2} \pm \cancel{\frac{5}{5}} \sin \theta \times \frac{1}{2} = \frac{\cos \theta}{2} \pm \sin \theta \times \frac{1}{2}$$. 5. Factor out $$\frac{1}{2}$$: $$\frac{1}{2} (\cos \theta \pm \sin \theta)$$. 6. The simplified expression is: $$\boxed{\frac{1}{2} (\cos \theta \pm \sin \theta)}$$. This means the original expression simplifies to half the sum or difference of $$\cos \theta$$ and $$\sin \theta$$.