Since correlations are cosines, for three random variables such that $\operatorname{corr}(A, B)=b$, $\operatorname{corr}(A, C)=c$ and $\operatorname{corr}(B, C)=a$, one must have $$ a \geqslant b c-\sqrt{1-b^{2}} \sqrt{1-c^{2}} $$ How can I prove the above?
I am fine up to $1-a^2+b(b-ca)-c(ba-c) \ge 0$ using Sylvestor's criterion.