Solve `cos^6(pi/16)+cos^6((3pi)/16)+cos^6((5pi)/16)+cos^6((7pi)/16)`

Explanation

To solve the expression \( \cos^6\left(\frac{\pi}{16}\right) + \cos^6\left(\frac{3\pi}{16}\right) + \cos^6\left(\frac{5\pi}{16}\right) + \cos^6\left(\frac{7\pi}{16}\right) \), we will follow these steps:

### Step 1: Rewrite the Cosine Terms
Using the identity \( \cos\left(\frac{\pi}{2} - x\right) = \sin(x) \), we can rewrite some of the cosine terms:
- \( \cos\left(\frac{5\pi}{16}\right) = \sin\left(\frac{3\pi}{16}\right) \)
- \( \cos\left(\frac{7\pi}{16}\right) = \sin\left(\frac{\pi}{16}\right) \)

Thus, we can express the original equation as:
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