Find the exact value of $\cos \frac{5 \pi}{12}$ by using a sum or difference formula.

Step 1 ：First, we need to express \(\frac{5 \pi}{12}\) as a sum or difference of angles that we know the cosine of. We know the cosine of \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\), and \(\frac{\pi}{4} + \frac{\pi}{3} = \frac{5 \pi}{12}\).

Step 2 ：So, we can write \(\cos \frac{5 \pi}{12} = \cos (\frac{\pi}{4} + \frac{\pi}{3})\).

Step 3 ：Next, we use the cosine sum formula, which states that \(\cos (a + b) = \cos a \cos b - \sin a \sin b\). Applying this to our equation gives us \(\cos \frac{5 \pi}{12} = \cos \frac{\pi}{4} \cos \frac{\pi}{3} - \sin \frac{\pi}{4} \sin \frac{\pi}{3}\).

Step 4 ：We know that \(\cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}\), \(\cos \frac{\pi}{3} = \frac{1}{2}\), \(\sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}\), and \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\). Substituting these values in gives us \(\cos \frac{5 \pi}{12} = \frac{1}{\sqrt{2}} \cdot \frac{1}{2} - \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}\).

Step 5 ：Simplifying this gives us \(\cos \frac{5 \pi}{12} = \frac{1}{2 \sqrt{2}} - \frac{\sqrt{3}}{2 \sqrt{2}}\).

Step 6 ：Finally, we can simplify this further by rationalizing the denominator, which gives us \(\cos \frac{5 \pi}{12} = \frac{\sqrt{2} - \sqrt{6}}{4}\).

Step 7 ：So, the exact value of \(\cos \frac{5 \pi}{12}\) is \(\boxed{\frac{\sqrt{2} - \sqrt{6}}{4}}\).