Given that cosx=2/3, find the exact value of cos4x

To find the exact value of \(\cos(4x)\) given that \(\cos(x) = \frac{2}{3}\), we can use the double-angle identities in trigonometry.

### Step 1: Find \(\cos(2x)\) using the double-angle identity:
\[
\cos(2x) = 2\cos^2(x) - 1
\]
Substitute \(\cos(x) = \frac{2}{3}\):
\[
\cos(2x) = 2\left(\frac{2}{3}\right)^2 - 1 = 2\left(\frac{4}{9}\right) - 1 = \frac{8}{9} - 1 = \frac{8}{9} - \frac{9}{9} = -\frac{1}{9}
\]

### Step 2: Find \(\cos(4x)\) using the identity for \(\cos(2x)\):
\[
\cos(4x) = 2\cos^2(2x) - 1
\]
Substitute \(\cos(2x) = -\frac{1}{9}\):
\[
\cos(4x) = 2\left(-\frac{1}{9}\right)^2 - 1 = 2\left(\frac{1}{81}\right) - 1 = \frac{2}{81} - 1 = \frac{2}{81} - \frac{81}{81} = -\frac{79}{81}
\]

### Final Answer:
\[
\cos(4x) = -\frac{79}{81}
\]

Would you like more details on these steps or do you have any questions? Here are some related questions you might find helpful:

1. How do you derive the double-angle identity for \(\cos(2x)\)?
2. What is the significance of the double-angle identities in trigonometry?
3. How can you find \(\sin(2x)\) given \(\cos(x)\) and \(\sin(x)\)?
4. What are the other trigonometric identities involving \(\cos(x)\) and \(\sin(x)\)?
5. How can you solve for \(\cos(3x)\) given \(\cos(x)\)?

**Tip:** Double-angle identities are useful for simplifying expressions and solving equations involving trigonometric functions.

Given that cosx=2/3, find the exact value of cos4x.

Discover how to find the exact value of cos(4x) when cos(x) = 2/3 using double-angle identities in trigonometry. This step-by-step solution explores key trigonometric formulas and double-angle theorems, suitable for high school students in Grades 10-12.

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