We're doing Basic Mathematics for Introductory level Physics in class, and my prof. illustrated a proof of the Law of Cosines. This he did by considering the following triangle and making use of the Pythagorean Theorem.
Proposed Proof:
Let $AB=c$, $AC=b$, $BC=a$. $\angle ADC$ by construction is a right angle, so the Pythagorean Theorem applies here. Also $AD=b\cos\theta$ and $CD=b\sin\theta$ where $\theta$ is the same as $\angle CAD$. Using Pythagorean Theorem we have: $(c+b\cos\theta)^2+(b\sin\theta)^2=c^2$. All that is to be done now is to rewrite $\theta=180^{\circ}-A$ and the result follows.
However is this proof not circular because it assumes the Pythagorean Theorem being true which itself is a special case of the Law of Cosines at $\theta=90^{\circ}$? Is this proof valid?
It is not circular because the Pythagorean theorem can be verified separately (by God knows how many methods of your choice: there are books full of proofs just for the Pythagorean theorem).
It might be easier to think of the law of cosines, not as permitting a special case of the Pythagorean theorem, so much as it is a generalization of the Pythagorean theorem to triangles that aren't necessarily right.
The law might subsume the Pythagorean theorem, in other words, but that doesn't mean the Pythagorean theorem itself can't be validated in other ways.