Let's break this down step by step:

The given equation is:

$\cos 4\theta = 1 - 8\sin^2 \theta \cos^2 \theta$

You are tasked with finding the exact value of:

$\sin^2 \left( \frac{\pi}{24} \right) \cos^2 \left( \frac{\pi}{24} \right)$

Step 1: Simplify the product $\sin^2 \left( \frac{\pi}{24} \right) \cos^2 \left( \frac{\pi}{24} \right)$

We can use the identity:

$\sin^2 x \cos^2 x = \frac{1}{4} \sin^2 (2x)$

So,

$\sin^2 \left( \frac{\pi}{24} \right) \cos^2 \left( \frac{\pi}{24} \right) = \frac{1}{4} \sin^2 \left( \frac{\pi}{12} \right)$

Step 2: Evaluate $\sin^2 \left( \frac{\pi}{12} \right)$

$\frac{\pi}{12}$ is a common angle. Using exact trigonometric values, we know:

$\sin \left( \frac{\pi}{12} \right) = \sin 15^\circ = \frac{\sqrt{6} - \sqrt{2}}{4}$

Now square it:

$\sin^2 \left( \frac{\pi}{12} \right) = \left( \frac{\sqrt{6} - \sqrt{2}}{4} \right)^2 = \frac{6 - 2\sqrt{12} + 2}{16} = \frac{8 - 4\sqrt{3}}{16} = \frac{2 - \sqrt{3}}{4}$

Step 3: Substitute this into the expression

Now substitute this into the equation:

$\sin^2 \left( \frac{\pi}{24} \right) \cos^2 \left( \frac{\pi}{24} \right) = \frac{1}{4} \times \frac{2 - \sqrt{3}}{4} = \frac{2 - \sqrt{3}}{16}$

Thus, the exact value is:

$\boxed{\frac{2 - \sqrt{3}}{16}}$

Would you like further clarification on any of the steps? Here are some related questions for deeper understanding:

1. How do trigonometric identities help in simplifying expressions?
2. What are the exact values of sine and cosine for other common angles?
3. How can you verify the exact values of trigonometric functions?
4. Why is the double angle identity useful in this context?
5. Can the identity $\sin^2 x \cos^2 x = \frac{1}{4} \sin^2(2x)$ be applied to other problems?

Tip: Memorizing key trigonometric values like $\sin 15^\circ$ and $\cos 15^\circ$ can be very helpful for problems like this.