# Why is the correlation coefficient between -1 and 1?

There are quite a few variants of the Cauchy-Schwarz inequality and I'm saying this because there is one related to statistics as well and it goes like this

There are quite a few variants of the Cauchy-Schwarz inequality and I'm saying this because there is one related to statistics as well and it goes like this:
$\operatorname{Cov}^{2}\left(X,Y\right) \le \operatorname{Var}\left(X\right)\operatorname{Var}\left(Y\right)$
It follows from this Cauchy-Schwarz inequality that the correlation coefficient is between -1 and 1.
$-\sqrt{\operatorname{Var}\left(X\right)\operatorname{Var}\left(Y\right)} \le \operatorname{Cov}\left(X,Y\right) \le \sqrt{\operatorname{Var}\left(X\right)\operatorname{Var}\left(Y\right)}$
Therefore,
$-1 \le \frac{\operatorname{Cov}\left(X,Y\right)}{\sqrt{\operatorname{Var}\left(X\right)\operatorname{Var}\left(Y\right)}}\le1$
The term in the middle is exactly the correlation coefficient!
Actually the covariance is the inner product between two random variables and the standard deviation is the norm of a random variable. If we denote the inner product $<X,Y>$ and the norm $|X|$, then the usual Cauchy-Schwarz inequality still holds:
$<X,Y>^{2}\le |X|^{2}|Y|^{2}.$
But then the inner product is also defined as follows:
$<X,Y>=|X||Y|\cos\left(\theta\right)$
The correlation coefficient is in fact the cosine of the angle between two variables:
$\operatorname{Corr}\left(X,Y\right) =\frac{<X,Y>}{|X||Y|}=\cos\left(\theta\right)$
It's obvious that cosine ranges only from -1 to 1, right?

Why is the correlation coefficient between -1 and 1?