Find the least positive value of $\theta$.
\[
\cos \left(4 \theta-8^{\circ}\right) \sec \left(3 \theta-1^{\circ}\right)=1
\]
\[
\theta=
\]

Step 1 ：Given the equation \(\cos \left(4 \theta-8^{\circ}\right) \sec \left(3 \theta-1^{\circ}\right)=1\)

Step 2 ：We know that \(\sec(x)\) is the reciprocal of \(\cos(x)\). So, we can rewrite the equation as \(\cos \left(4 \theta-8^{\circ}\right) \cdot \frac{1}{\cos \left(3 \theta-1^{\circ}\right)}=1\)

Step 3 ：This simplifies to \(\cos \left(4 \theta-8^{\circ}\right) = \cos \left(3 \theta-1^{\circ}\right)\)

Step 4 ：We can solve this equation for \(\theta\) by setting the two expressions equal to each other and solving for \(\theta\)

Step 5 ：The solutions for \(\theta\) are \(7.00000000000000, -101.571428571429, 52.7142857142857, 104.142857142857, -50.1428571428571, 155.571428571429, -153.000000000000, 1.28571428571429\) degrees

Step 6 ：The least positive value of \(\theta\) is \(1.28571428571429\) degrees

Step 7 ：Final Answer: The least positive value of \(\theta\) is \(\boxed{1.28571428571429}\) degrees