To find the exact value of $\sec\left(\frac{11\pi}{8}\right)$ using a half-angle formula, we can follow these steps:

1. Identify the Reference Angle: The angle $\frac{11\pi}{8}$ is in the third quadrant, where the secant function is negative. Its reference angle is:

   $\pi - \frac{11\pi}{8} = \frac{8\pi}{8} - \frac{11\pi}{8} = -\frac{3\pi}{8}$

2. Use the Half-Angle Formula for Cosine: The secant function is the reciprocal of the cosine function:

   $\sec\left(\theta\right) = \frac{1}{\cos\left(\theta\right)}$

   The half-angle formula for cosine is:

   $\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}$

   Here, we need to use $\theta = \frac{11\pi}{4}$ (since $\frac{11\pi}{8} = \frac{1}{2} \cdot \frac{11\pi}{4}$).

3. Calculate $\cos\left(\frac{11\pi}{4}\right)$: This angle can be simplified:

   $\frac{11\pi}{4} = 2\pi + \frac{3\pi}{4}$

   Thus,

   $\cos\left(\frac{11\pi}{4}\right) = \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

4. Apply the Half-Angle Formula:

   $\cos\left(\frac{11\pi}{8}\right) = -\sqrt{\frac{1 + \left(-\frac{\sqrt{2}}{2}\right)}{2}} = -\sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = -\sqrt{\frac{2 - \sqrt{2}}{4}} = -\frac{\sqrt{2 - \sqrt{2}}}{2}$

5. Calculate Secant:

   $\sec\left(\frac{11\pi}{8}\right) = \frac{1}{\cos\left(\frac{11\pi}{8}\right)} = -\frac{2}{\sqrt{2 - \sqrt{2}}}$

6. Rationalize the Denominator:

   Multiply numerator and denominator by $\sqrt{2 + \sqrt{2}}$:

   $\sec\left(\frac{11\pi}{8}\right) = -\frac{2\sqrt{2 + \sqrt{2}}}{\sqrt{(2 - \sqrt{2})(2 + \sqrt{2})}} = -\frac{2\sqrt{2 + \sqrt{2}}}{\sqrt{4 - 2}} = -\frac{2\sqrt{2 + \sqrt{2}}}{\sqrt{2}} = -\sqrt{2(2 + \sqrt{2})}$

Thus, the final answer is:

$\sec\left(\frac{11\pi}{8}\right) = -\sqrt{2(2 + \sqrt{2})}$

Would you like further details on any specific steps or concepts? Here are some related questions you might consider:

1. What are the properties of secant and cosine functions?
2. How do half-angle formulas apply to different trigonometric functions?
3. Can you explain the significance of reference angles in trigonometry?
4. How do you rationalize denominators in expressions involving radicals?
5. What are other applications of half-angle formulas in trigonometry?

Tip: Always remember to consider the sign of trigonometric functions based on the quadrant in which the angle lies.