\(\cos ^3\left(\frac{\pi}{8}\right) \cos \left(\frac{3 \pi}{8}\right)+\sin ^3\left(\frac{\pi}{8}\right) \sin \left(\frac{3 \pi}{8}\right)=\)

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:
\(\frac{\sqrt{2}}{4}\)