To express \(\cos(8x)\) as a function of \(\sin(x)\), we can use trigonometric identities, specifically the double-angle and multiple-angle identities. Here's the step-by-step process:

1. **Use the double-angle identity:**
   \[
   \cos(2\theta) = 2\cos^2(\theta) - 1
   \]
   We can use this identity iteratively.

2. **Express \(\cos(2x)\):**
   \[
   \cos(2x) = 2\cos^2(x) - 1
   \]

3. **Express \(\cos(4x)\):**
   \[
   \cos(4x) = 2\cos^2(2x) - 1 = 2(2\cos^2(x) - 1)^2 - 1
   \]

4. **Express \(\cos(8x)\):**
   \[
   \cos(8x) = 2\cos^2(4x) - 1
   \]
   Substitute \(\cos(4x)\) from the previous step:
   \[
   \cos(8x) = 2\left[2(2\cos^2(x) - 1)^2 - 1\right]^2 - 1
   \]

5. **Express \(\cos^2(x)\) as a function of \(\sin(x)\):**
   \[
   \cos^2(x) = 1 - \sin^2(x)
   \]
   Substitute \(\cos^2(x) = 1 - \sin^2(x)\) into the expression for \(\cos(8x)\):

   \[
   \cos(8x) = 2\left[2(2(1 - \sin^2(x)) - 1)^2 - 1\right]^2 - 1
   \]

This is the expression for \(\cos(8x)\) as a function of \(\sin(x)\), but it's quite complex due to the multiple iterations of identities. You can further expand and simplify it if needed.

Would you like a detailed expansion of this result or have any questions?

### Related Questions:
1. How can we express \(\cos(16x)\) as a function of \(\sin(x)\)?
2. What is the simplified form of \(\cos(4x)\) in terms of \(\sin(x)\)?
3. How would you express \(\sin(8x)\) as a function of \(\sin(x)\)?
4. Can we use a similar approach to express \(\cos(3x)\) in terms of \(\sin(x)\)?
5. How does expressing \(\cos\) in terms of \(\sin\) help in solving trigonometric equations?

**Tip:** For easier calculations, remember that \(\cos^2(x) = 1 - \sin^2(x)\) is a key identity that frequently simplifies trigonometric expressions.

Learn how to express cos(8x) as a function of sin(x) using trigonometric identities. This detailed step-by-step solution uses the double-angle and multiple-angle identities to derive the expression. Suitable for advanced high school students and anyone interested in trigonometry.

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