Consider three random variables $X,Y,Z$ with correlation matrix $$\begin{bmatrix}1& \rho_{x,y} & \rho_{x,z} \\ \rho_{x,y} & 1 & \rho_{y,z} \\ \rho_{x,z} & \rho_{y,z} & 1 \end{bmatrix} $$ Given $\rho_{x,y}, \rho_{x,z}$, what are the possible values for $\rho_{y,z}$?
I know that the correlation matrix should be positive semi-definite. This gives us the following condition: $$1-\rho_{y,z}^2 - \rho_{x,y}(\rho_{x,y} - \rho_{x,z}\rho_{y,z}) + \rho_{x,z}(\rho_{x,y} \rho_{y,z} - \rho_{x,z}))\geq 0 $$I can find the bound from the above condition. However, I saw another bound derived using cosine-principle (see Correlation between three variables question). It goes like this: $$\rho_{y,z} \leq \rho_{x,y}\rho_{x,z} - \sqrt{1-\rho_{x,y}^2}\sqrt{1-\rho_{x,z}^2} $$ I want to understand if it is possible to obtain the second bound from the first bound.
