Exercises: Prove the Laws of Sines and Cosines

Complete the final step in the Law of Cosines proof. After expanding the distance formula and regrouping, we have: $c^2 = a^2(\cos^2 C + \sin^2 C) + b^2 - 2ab\cos C$. Applying the Pythagorean identity $\sin^2 C + \cos^2 C = 1$ gives: $c^2 =$   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲