I'm wondering where the following equality came from: $$ \langle x , y \rangle = \|x \| \| y \| \cos \theta$$
where the thing on the LHS is the inner product and $\|\cdot\|$ is the norm induced by $\langle \cdot, \cdot \rangle$. Do we need the Cauchy Schwarz inequality to prove this? I'm asking because I'm reading my notes and there is a proof of the C. S. - inequality. It's fairly short but longer than the following:
Claim: $|\langle x,y \rangle | \leq \|x \| \|y \|$
Proof: Since $\langle x , y \rangle = \|x \| \| y \| \cos \theta$ we have $-\|x \| \| y \| \leq \langle x , y \rangle \leq \|x \| \| y \|$.