To solve the expression $\cos ^3\left(\frac{\pi}{8}\right) \cos \left(\frac{3 \pi}{8}\right)+\sin ^3\left(\frac{\pi}{8}\right) \sin \left(\frac{3 \pi}{8}\right)$, we will follow these steps:
Step 1: Rewrite $\cos \left(\frac{3 \pi}{8}\right)$ and $\sin \left(\frac{3 \pi}{8}\right)$
Using the identity $\cos \left(\frac{3 \pi}{8}\right)=\sin \left(\frac{\pi}{2}-\frac{3 \pi}{8}\right)=\sin \left(\frac{\pi}{8}\right)$ and $\sin \left(\frac{3 \pi}{8}\right)=\cos \left(\frac{\pi}{2}-\frac{3 \pi}{8}\right)=\cos \left(\frac{\pi}{8}\right)$, we can rewrite the expression:
$\cos ^3\left(\frac{\pi}{8}\right) \sin \left(\frac{\pi}{8}\right)+\sin ^3\left(\frac{\pi}{8}\right) \cos \left(\frac{\pi}{8}\right)$
Step 2: Factor the expression
Now we can factor out $\cos \left(\frac{\pi}{8}\right) \sin \left(\frac{\pi}{8}\right)$ :
$\cos \left(\frac{\pi}{8}\right) \sin \left(\frac{\pi}{8}\right)\left(\cos ^2\left(\frac{\pi}{8}\right)+\sin ^2\left(\frac{\pi}{8}\right)\right)$
Step 3: Simplify using Pythagorean identity
Using the Pythagorean identity $\cos ^2 A+\sin ^2 A=1$ :
Step 3: Simplify using Pythagorean identity
Using the Pythagorean identity $\cos ^2 A+\sin ^2 A=1$ :
$\cos \left(\frac{\pi}{8}\right) \sin \left(\frac{\pi}{8}\right) \cdot 1=\cos \left(\frac{\pi}{8}\right) \sin \left(\frac{\pi}{8}\right)$
Step 4: Use the double angle formula
We can use the double angle formula for sine, which states that $\sin (2 A)=2 \sin (A) \cos (A)$ :
$\sin \left(\frac{\pi}{4}\right)=2 \sin \left(\frac{\pi}{8}\right) \cos \left(\frac{\pi}{8}\right)$
Since $\sin \left(\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}$ :
$\cos \left(\frac{\pi}{8}\right) \sin \left(\frac{\pi}{8}\right)=\frac{1}{2} \sin \left(\frac{\pi}{4}\right)=\frac{1}{2} \cdot \frac{\sqrt{2}}{2}=\frac{\sqrt{2}}{4}$
Final Answer
Thus, the value of the expression is:
$\frac{\sqrt{2}}{4}$