Use a sum or difference formula to find the exact value of the following.
$$\sin \frac{29\pi }{42}\cos \frac{\pi }{7}+\cos \frac{29\pi }{42}\sin \frac{\pi }{7}$$

Step 1 ：The given expression is in the form of the sum of products of sine and cosine of two angles. This is similar to the sine of sum of two angles formula, which is given by: $\sin (A+B)=\sin A\cos B+\cos A\sin B$

Step 2 ：So, we can use this formula to simplify the given expression. The two angles in the given expression are $\frac{29\pi }{42}$ and $\frac{\pi }{7}$.

Step 3 ：Let A = $\frac{29\pi }{42}$ and B = $\frac{\pi }{7}$

Step 4 ：Substitute A and B into the formula, we get $\sin (A+B)=\sin A\cos B+\cos A\sin B$

Step 5 ：The result from the calculation is $\frac{1}{2}$. This means that the exact value of the given expression is $\frac{1}{2}$.

Step 6 ：Final Answer: The exact value of the given expression is $\boxed{{\displaystyle \frac{1}{2}}}$.