Let's evaluate the expression:

$\sin^6 \frac{\pi}{16} + \sin^6 \frac{3\pi}{16} + \sin^6 \frac{5\pi}{16} + \sin^6 \frac{7\pi}{16}$

Step 1: Simplify Using Symmetry

The angles in the sine terms can be paired using symmetry properties:

• $\sin\left(\frac{\pi}{16}\right) = \sin\left(\frac{7\pi}{16}\right)$
• $\sin\left(\frac{3\pi}{16}\right) = \sin\left(\frac{5\pi}{16}\right)$

Thus, the expression can be rewritten as:

$2\left(\sin^6 \frac{\pi}{16} + \sin^6 \frac{3\pi}{16}\right)$

Step 2: Use the Identity for Sine Power

Use the identity: $\sin^6 x = \frac{1}{32} \left(10 - 15 \cos(2x) + 6 \cos(4x) - \cos(6x)\right)$ for each sine term.

Step 3: Compute the Values

Substitute $\frac{\pi}{16}$ and $\frac{3\pi}{16}$ into the identity:

$\sin^6 \frac{\pi}{16} = \frac{1}{32} \left(10 - 15 \cos\left(\frac{\pi}{8}\right) + 6 \cos\left(\frac{\pi}{4}\right) - \cos\left(\frac{3\pi}{8}\right)\right)$ $\sin^6 \frac{3\pi}{16} = \frac{1}{32} \left(10 - 15 \cos\left(\frac{3\pi}{8}\right) + 6 \cos\left(\frac{3\pi}{4}\right) - \cos\left(\frac{9\pi}{8}\right)\right)$

Final Result

After evaluating these and summing them up, the simplified final result of the expression is:

$\boxed{\frac{5}{8}}$

This exact value represents the sum of the original sine powers.

Would you like more details on how these identities were used or need help with other questions?

Here are some related questions you might find interesting:

1. How do you derive the identity for $\sin^6 x$?
2. Can similar identities be used for cosines?
3. How do you generalize sums of powers of sine functions?
4. How does symmetry simplify trigonometric sums?
5. What other trigonometric identities simplify power sums?

Tip: Trigonometric identities are powerful tools for simplifying complex expressions—knowing a few key ones can save a lot of time!