1. The problem is to simplify the expression $\cos a - \cos b$. 2. Recall the cosine difference identity: $$\cos a - \cos b = -2 \sin \left( \frac{a+b}{2} \right) \sin \left( \frac{a-b}{2} \right)$$ 3. Substitute the expression into this identity for simplification: Thus, $$\cos a - \cos b = -2 \sin \left( \frac{a+b}{2} \right) \sin \left( \frac{a-b}{2} \right)$$ 4. This is the simplified form of the difference of cosines using trigonometric identities. Final answer: $$\boxed{\cos a - \cos b = -2 \sin \left( \frac{a+b}{2} \right) \sin \left( \frac{a-b}{2} \right)}$$