If $$L = \sin^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right)$$ and $$M = \cos^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right)$$

Step 1: Calculate the Sum (L + M)

First, let us add the two given expressions together.

$$L = \sin^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right)$$
$$M = \cos^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right)$$

Adding them yields:

$$L + M = \left[\sin^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right)\right] + \left[\cos^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right)\right]$$

Group the terms with the same angles:

$$L + M = \left[\sin^2\left(\frac{\pi}{16}\right) + \cos^2\left(\frac{\pi}{16}\right)\right] - 2\sin^2\left(\frac{\pi}{8}\right)$$

Apply the fundamental identity $$\sin^2\theta + \cos^2\theta = 1$$ to the first bracket:

$$L + M = 1 - 2\sin^2\left(\frac{\pi}{8}\right)$$

Now, apply the double angle identity $$\cos(2\theta) = 1 - 2\sin^2\theta$$:

$$L + M = \cos\left(2 \times \frac{\pi}{8}\right)$$
$$L + M = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$$

Step 2: Calculate the Difference (L - M)

Next, let us subtract $$M$$ from $$L$$.

$$L - M = \left[\sin^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right)\right] - \left[\cos^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right)\right]$$

The $$\sin^2\left(\frac{\pi}{8}\right)$$ terms cancel out:

$$L - M = \sin^2\left(\frac{\pi}{16}\right) - \cos^2\left(\frac{\pi}{16}\right)$$

Factor out a negative sign to match the standard double angle formula:

$$L - M = -\left[\cos^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{16}\right)\right]$$

Apply the double angle identity $$\cos(2\theta) = \cos^2\theta - \sin^2\theta$$:

$$L - M = -\cos\left(2 \times \frac{\pi}{16}\right)$$
$$L - M = -\cos\left(\frac{\pi}{8}\right)$$

Step 3: Solve for M

We now have a simple system of two equations:

1. $$L + M = \frac{1}{\sqrt{2}}$$
2. $$L - M = -\cos\left(\frac{\pi}{8}\right)$$

To isolate $$M$$, subtract equation (2) from equation (1):

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

Divide the entire equation by 2 to find the final value of $$M$$:

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

Hence, the answer is option D.