Analyzing a signal that contains frequency content at Fs/2 doesn't seem to work unless there is a phase shift
the discrete function $$x_q[n]=\sin(\pi n)$$ is always zero for all of the integers $n$.

the discrete function $$x_i[n]=\cos(\pi n)$$ is always $(-1)^n$ for integer $n$.

so this general sinusoid at Nyquist that has a phase term:

$$\begin{align}
x[n] &amp;= A \cos(\pi n + \theta) \\
     &amp;= A \big( \cos(\pi n) \cos(\theta) - \sin(\pi n) \sin(\theta) \big) \\
     &amp;= \big(A\cos(\theta)\big) \cos(\pi n)  +  \big(-A\sin(\theta)\big) \sin(\pi n) \\
     &amp;= \big(A\cos(\theta)\big) (-1)^n  +  \big(-A\sin(\theta)\big) \cdot 0 \\
     &amp;= B (-1)^n \\
\end{align}$$

So now, how do you tell the difference between a sampled sinusoid that had amplitude $A$ and a phase angle $\theta$ and another sampled sinusoid that has amplitude $B \triangleq\big(A\cos(\theta)\big)$ and no phase shift from the cosine?