How can you prove that the squares of the expected values of the three components of spin sum to 1?

I am working through Leonard Susskind’s The Theoretical Minimum: Quantum Mechanics. In this book a statement called the "spin-polarization principle" is introduced, which essentially states that:

For any state $|A rangle$, there exists a direction vector $hat{n}$ such that $vec{sigma} cdot hat{n} , |A rangle = |Arangle$.

(Here $sigma$ is used for spin operators – I’ve seen $S_n, S_x, S_y, S_z$ used elsewhere).

I understand this to mean that for any state there always exists a direction that a spin-measuring apparatus can be oriented in such that it will measure the spin as $+1$ with $100%$ certainty. We can therefore write that the expectation value of this observable is $1$:

$$langle vec{sigma} cdot hat{n} rangle = 1.$$

We have that $vec{sigma} cdot hat{n} = n_x sigma_x + n_y sigma_y + n_z sigma_z$ where $n_x, n_y, n_z $ are the components of $hat{n}$.

The book then states that "the expectation value of the perpendicular components of $sigma$ are zero in the state $|Arangle$" and then states that it "follows" from this that

$$langle sigma_x rangle ^2 + langle sigma_y rangle ^2 + langle sigma_z rangle ^2 = 1.$$

I don’t understand what the book means by this though, or how you deduce that the sum of the squares of the expected values of the spin components is 1.

I think it might be that the "perpendicular components" being $0$ refers to the spin component measured in a direction perpendicular to $hat{n}$ in $3$D space (because if a $+1$ spin is prepared along $hat{n}$ then the expected value of the spin measurement perpendicular to $hat{n}$ (along a vector $hat{m}$) is $hat{n} cdot hat{m} = 0$.

We also can show that

$$langle vec{sigma} cdot hat{n} rangle = langle A | vec{sigma} cdot hat{n} | A rangle = n_x langle sigma_x rangle + n_y langle sigma_y rangle + n_z langle sigma_z rangle,$$

which is the closest that I’ve got to showing that the sum of the squares of the expected values of the spin components is $1$.

What does the book mean by this statement, and how do we deduce that

$$langle sigma_x rangle ^2 + langle sigma_y rangle ^2 + langle sigma_z rangle ^2 = 1?$$