Let's solve this step by step.

Given:

1. $\sec e \cdot \cos x = \sqrt{2}$
2. $\tan e \cdot \cot a = \sqrt{3}$

Step 1: Understand the First Equation

The first equation is $\sec e \cdot \cos x = \sqrt{2}$.

Recall that: $\sec \theta = \frac{1}{\cos \theta}$ So, the equation becomes: $\frac{1}{\cos e} \cdot \cos x = \sqrt{2}$

Simplify: $\frac{\cos x}{\cos e} = \sqrt{2}$

So, the ratio between $\cos x$ and $\cos e$ is $\sqrt{2}$.

Step 2: Understand the Second Equation

The second equation is $\tan e \cdot \cot a = \sqrt{3}$.

Recall that: $\tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta}$ So, the equation becomes: $\frac{\sin e}{\cos e} \cdot \frac{\cos a}{\sin a} = \sqrt{3}$ Simplifying: $\frac{\sin e \cdot \cos a}{\cos e \cdot \sin a} = \sqrt{3}$ This can be rewritten as: $\frac{\sin e}{\sin a} \cdot \frac{\cos a}{\cos e} = \sqrt{3}$

Step 3: Solve for $\tan e$ and $\tan d$

To solve for $\tan e$ and $\tan d$, we'll need more information or assumptions about $e$ and $d$.

Do you have any additional conditions or information for the angles involved? Or would you like me to further clarify or solve based on assumptions?