Let $x=\cos 36^{\circ} - \cos 72^{\circ}$. Then $x$ equals

$ \textbf{(A)}\ \frac{1}{3} \qquad\textbf{(B)}\ \frac{1}{2} \qquad\textbf{(C)}\ 3-\sqrt{6} \qquad\textbf{(D)}\ 2\sqrt{3}-3 \qquad\textbf{(E)}\ \text{none of these} $
1. We start with the given expression \( x = \cos 36^\circ - \cos 72^\circ \).

2. We use the double angle formulas for cosine:
   \[
   \cos 36^\circ = 1 - 2 \sin^2 18^\circ
   \]
   and
   \[
   \cos 72^\circ = 2 \cos^2 36^\circ - 1.
   \]

3. We need to express \(\cos 36^\circ\) and \(\cos 72^\circ\) in a form that allows us to simplify \( x \). First, we use the identity for \(\cos 72^\circ\):
   \[
   \cos 72^\circ = 2 \cos^2 36^\circ - 1.
   \]

4. Let \( y = \cos 36^\circ \). Then:
   \[
   \cos 72^\circ = 2y^2 - 1.
   \]

5. Substitute these into the expression for \( x \):
   \[
   x = y - (2y^2 - 1).
   \]

6. Simplify the expression:
   \[
   x = y - 2y^2 + 1.
   \]

7. We need to find the value of \( y = \cos 36^\circ \). Using the known value:
   \[
   \cos 36^\circ = \frac{\sqrt{5} + 1}{4}.
   \]

8. Substitute \( y = \frac{\sqrt{5} + 1}{4} \) into the expression for \( x \):
   \[
   x = \frac{\sqrt{5} + 1}{4} - 2 \left( \frac{\sqrt{5} + 1}{4} \right)^2 + 1.
   \]

9. Calculate \( \left( \frac{\sqrt{5} + 1}{4} \right)^2 \):
   \[
   \left( \frac{\sqrt{5} + 1}{4} \right)^2 = \frac{(\sqrt{5} + 1)^2}{16} = \frac{5 + 2\sqrt{5} + 1}{16} = \frac{6 + 2\sqrt{5}}{16} = \frac{3 + \sqrt{5}}{8}.
   \]

10. Substitute back into the expression for \( x \):
    \[
    x = \frac{\sqrt{5} + 1}{4} - 2 \cdot \frac{3 + \sqrt{5}}{8} + 1.
    \]

11. Simplify the terms:
    \[
    x = \frac{\sqrt{5} + 1}{4} - \frac{3 + \sqrt{5}}{4} + 1.
    \]

12. Combine the fractions:
    \[
    x = \frac{\sqrt{5} + 1 - 3 - \sqrt{5}}{4} + 1 = \frac{-2}{4} + 1 = -\frac{1}{2} + 1 = \frac{1}{2}.
    \]

13. Therefore, the value of \( x \) is:
    \[
    x = \frac{1}{2}.
    \]

The final answer is \(\boxed{\frac{1}{2}}\).