Step 1: Turn cosec inverse into sine inverse.
Since $\operatorname{cosec}^{-1}(\theta) = \sin^{-1}(1/\theta)$, the first term flips its fraction:
\[ \operatorname{cosec}^{-1}\!\left(\tfrac{\sqrt{x}+1}{\sqrt{x}-1}\right) = \sin^{-1}\!\left(\tfrac{\sqrt{x}-1}{\sqrt{x}+1}\right). \]
Step 2: Notice both terms share one argument.
Now $y = \sin^{-1}(u) + \cos^{-1}(u)$ where $u = \tfrac{\sqrt{x}-1}{\sqrt{x}+1}$.
Step 3: Apply the standard identity.
For any allowed $u$, $\sin^{-1}(u) + \cos^{-1}(u) = \tfrac{\pi}{2}$. So $y$ is just a constant.
Step 4: Differentiate.
The derivative of a constant is zero, so $\tfrac{dy}{dx} = 0$.
\[ \boxed{0} \]