1. Interpret the expressions

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

We will simplify both using $\sin^2 x=\frac{1-\cos 2x}{2}, \qquad \cos^2 x=\frac{1+\cos 2x}{2}.$

2. Find $L$

Using the identity for $\sin^2 x$, $\sin^2\left(\frac{\pi}{16}\right)=\frac{1-\cos\left(\frac\pi8\right)}{2}$ and $\sin^2\left(\frac{\pi}{8}\right)=\frac{1-\cos\left(\frac\pi4\right)}{2}.$

So, $L=\frac{1-\cos\left(\frac\pi8\right)}{2}-\frac{1-\cos\left(\frac\pi4\right)}{2}.$

This gives $L=\frac{-\cos\left(\frac\pi8\right)+\cos\left(\frac\pi4\right)}{2}.$

Since $\cos\left(\frac\pi4\right)=\frac{1}{\sqrt2},$ we get $L=\frac{1}{2\sqrt2}-\frac{1}{2}\cos\left(\frac\pi8\right).$

So, $L=\frac{1}{2\sqrt2}-\frac{1}{2}\cos\frac\pi8.$

Now compare with options:

• Option A says $L=-\frac{1}{2\sqrt2}+\frac12\cos\frac\pi8,$ which is the negative of our result, so false.

• Option D says $L=\frac{1}{4\sqrt2}-\frac14\cos\frac\pi8,$ which is half of our result, so false.

3. Find $M$

Using the identities, $\cos^2\left(\frac{\pi}{16}\right)=\frac{1+\cos\left(\frac\pi8\right)}{2}$ and $\sin^2\left(\frac{\pi}{8}\right)=\frac{1-\cos\left(\frac\pi4\right)}{2}.$

Therefore, $M=\frac{1+\cos\left(\frac\pi8\right)}{2}-\frac{1-\cos\left(\frac\pi4\right)}{2}.$

Simplifying, $M=\frac{\cos\left(\frac\pi8\right)+\cos\left(\frac\pi4\right)}{2}.$

Again using $\cos\left(\frac\pi4\right)=\frac{1}{\sqrt2},$ we get $M=\frac12\cos\frac\pi8+\frac{1}{2\sqrt2}.$

So, $M=\frac{1}{2\sqrt2}+\frac12\cos\frac\pi8.$

Now compare with options:

• Option B says $M=\frac{1}{2\sqrt2}+\frac12\cos\frac\pi8,$ so true.

• Option C says $M=\frac{1}{4\sqrt2}+\frac14\cos\frac\pi8,$ again half of the correct value, so false.

4. Conclusion

Only Option B is correct.