Step 1: Identify standard trigonometric values

From standard trigonometric table values, we have: $\cos 45° = \frac{1}{\sqrt{2}}$, $\tan 30° = \frac{1}{\sqrt{3}}$, and $\cot 60° = \frac{1}{\sqrt{3}}$.

$\cos 45° = \frac{1}{\sqrt{2}}, \quad \tan 30° = \frac{1}{\sqrt{3}}, \quad \cot 60° = \frac{1}{\sqrt{3}}$

Step 2: Substitute values into the given expression

Substitute these values into the expression $\frac{\cos 45°}{\tan 30° + \cot 60°}$: $\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{3}} + \frac{1}{\sqrt{3}}}$

$\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{3}} + \frac{1}{\sqrt{3}}}$

Step 3: Simplify the denominator

Add the terms in the denominator: $\frac{1}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{2}{\sqrt{3}}$. Thus, the expression becomes $\frac{\frac{1}{\sqrt{2}}}{\frac{2}{\sqrt{3}}}$

$\frac{\frac{1}{\sqrt{2}}}{\frac{2}{\sqrt{3}}} = \frac{\sqrt{3}}{2\sqrt{2}}$