Let's suppose we have an LTI system whose Laplace domain transfer function is: $$ F(s)=\frac{1}{s^2 + \frac{\omega_y}{Q_y}s + \omega_y^2} $$
Its input is the Coriolis force. Such force is experienced if you apply a force to a body along a direction and rotate it along another direction. In our case we move a body of mass $m$ along the $x$ axis: $$ x(t) = A_d\sin(\omega_dt) $$ As stated the coriolis force is experienced if you rotate that body, so we also rotate it along $z$ axis: $$ \Omega=\Omega_0\cos(\omega_st) $$
Coriolis acceleration is finally: $$ a_C = 2\Omega\dot{x}=2\omega_dA_d\Omega_0\cos(\omega_st)\cos(\omega_dt)= \omega_d\Omega_0A_d\Big[\cos[(\omega_d+\omega_s)t]+\cos[(\omega_d-\omega_s)t]\Big] $$ along y axis.
It happens that $F(s)$ is the transfer function that associates acceleration and position along the y axis, it actually is the TF of a mass, spring damper system.
Here comes the question: as you know, if you have a pure sinusoidal function in the time domain and you know its frequency you can compute the output just by calculating $F(s)|_{s=j\omega}$ where $j$ is the imaginary unit. In our case unfortunately this is not so, and my professor just writes: $$ \overline{y}=\omega_d\Omega_0A_d\cdot F[j(\omega_d\pm\omega_s)] $$ that's right, that's a $\pm$. We are only interested in the amplitude so the absence of any $\cos$ term is fine. The only hypotesis given till now is: $$ \text{HP:}\qquad \omega_s\ll\omega_d,\omega_y $$ He proceeds to make some simplifications using these hypotesis getting to: $$ \overline{y}\approx\frac{\Omega_0A_d}{2(\Delta_\omega \pm \omega_s)+j\Delta_y} $$ where: $$ \Delta_\omega \triangleq \omega_y - \omega_d\\ \Delta_y \triangleq \frac{\omega_y}{Q_y} $$ and this part is actually easy enough. The only way I can explain the $\pm$ thing is that if $\omega_d=\omega_y$ we are working on the resonance peak of $F(s)$ so for small $\omega_s$ (true) we have $F[j(\omega_d + \omega_s)]\approx F[j(\omega_d - \omega_s)]$, but he then analyzes two different cases, in one of them $\omega_d=\omega_y$ but in the oter $\omega_d\neq\omega_y$.
Is there any math deception I am missing or is my professor a bit inaccurate? He is usually quite rigorous, and quite good at omitting "easy" chunks of his reasoning.
$$F[j(\omega_d + \omega_s)]\approx F[j(\omega_d - \omega_s)]$$ You got it.
Try to insert for the case $\omega_d=\omega_y$, there will be exact cancellations and the final form should be slightly different.
In that case, you should reconsider the negligibility constraint.