Determine the value of the given trigonometric expression: $\cos \frac{35 \pi}{18} \cos \frac{5 \pi}{18}+\sin \frac{35 \pi}{18} \sin \frac{5 \pi}{18}$

Step 1 ：Determine the value of the given trigonometric expression: \( \cos \frac{35 \pi}{18} \cos \frac{5 \pi}{18}+\sin \frac{35 \pi}{18} \sin \frac{5 \pi}{18} \)

Step 2 ：The expression is in the form of the cosine of the sum of two angles. The formula for the cosine of the sum of two angles is \( \cos(A + B) = \cos A \cos B - \sin A \sin B \). However, in this case, the expression is \( \cos A \cos B + \sin A \sin B \), which is the cosine of the difference of two angles.

Step 3 ：The formula for the cosine of the difference of two angles is \( \cos(A - B) = \cos A \cos B + \sin A \sin B \). Therefore, the given expression can be simplified to \( \cos(\frac{35\pi}{18} - \frac{5\pi}{18}) \).

Step 4 ：Substitute \( A = 6.1086523819801535 \) and \( B = 0.8726646259971648 \) into the formula.

Step 5 ：The result of the calculation is approximately 0.5. This is the value of the given trigonometric expression.

Step 6 ：Final Answer: The value of the given trigonometric expression is \( \boxed{0.5} \)