To express \(\cos(8x)\) as a function of \(\sin(x)\), you can use trigonometric identities, specifically the double-angle identity for cosine:

\[
\cos(2\theta) = 1 - 2\sin^2(\theta)
\]

### Steps:

1. **Express \(\cos(8x)\) in terms of \(\cos(4x)\):**

   \[
   \cos(8x) = 2\cos^2(4x) - 1
   \]

2. **Express \(\cos(4x)\) in terms of \(\cos(2x)\):**

   \[
   \cos(4x) = 2\cos^2(2x) - 1
   \]

   Therefore,

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

3. **Express \(\cos(2x)\) in terms of \(\cos(x)\):**

   \[
   \cos(2x) = 2\cos^2(x) - 1
   \]

   So,

   \[
   \cos(4x) = 2(2(2\cos^2(x) - 1)^2 - 1)^2 - 1
   \]

4. **Express \(\cos(x)\) in terms of \(\sin(x)\):**

   \[
   \cos(x) = \sqrt{1 - \sin^2(x)}
   \]

   Substitute this into the previous expressions.

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

   Continuing this way for higher angles, the expression for \(\cos(8x)\) in terms of \(\sin(x)\) becomes very lengthy and complex. However, in principle, this is how you would systematically express \(\cos(8x)\) as a function of \(\sin(x)\).

Would you like to explore the detailed expansion, or have any other questions on this topic?  

Here are some relative questions:

1. How can \(\sin(8x)\) be expressed as a function of \(\sin(x)\)?
2. What is the general approach to express \(\cos(nx)\) as a function of \(\sin(x)\)?
3. Can you simplify trigonometric functions using sum and difference formulas?
4. How does expressing trigonometric functions in terms of others help in integration?
5. What are the common pitfalls when using trigonometric identities?

**Tip:** Using trigonometric identities often requires careful algebraic manipulation, so double-check each step to avoid errors.

Learn how to express cos(8x) as a function of sin(x) using trigonometric identities. Detailed step-by-step solution with algebraic manipulations and substitutions explained. Suitable for advanced high school students and beyond.

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