Let $\mathbb{R}^d,~d\geq 2,$ with standard inner product. Let $X,Y,Z$ be three non-zero vectors at $\mathbb{R}^d.$ Assume that $\theta$ is the angle between $X,Y$ and $\phi$ is the angle between $X,Z$. How can I determine the angle between $Y,Z$ in terms of $\phi,\theta$?
This is motivated by the following: consider the expression: $$\langle X,Y\rangle\langle X,Z\rangle = |X|^2|Y||Z|\cos\phi\cos\theta,$$ I would like to write the right side as some multiple of $\langle Y,Z\rangle$. I have the probably fake impression that the searched angle is something like $\frac{d\pi}{2} \pm (\varphi + \theta).$
Here's an example to consider. Let $n = 3$, and set $X = (1, 0, 0),\ Y = (0, 1, 0),\ \text{and}\ Z = (0, \cos t, \sin t)$ for $t$ a parameter. Then $X \cdot Y = 0 = X \cdot Z$, so $\theta = \phi = \pi/2$; note that these angles are independent of choice of $t$. But $Y \cdot Z = \cos t$, making $t$ the angle between $Y$ and $Z$. In other words, knowing $\theta$ and $\phi$ doesn't tell you about the third angle.