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The given expression is:

\[
\cos(a+b)\cos(a-b)
\]

with the conditions:

\[
\cos a = \frac{1}{3}, \quad \cos b = \frac{1}{4}.
\]

We will solve this using trigonometric identities and values for \(\cos(a)\) and \(\cos(b)\).

### Step-by-Step Solution:

1. **Expand the expression** using the product-to-sum identity for cosines:
   
   \[
   \cos(a+b)\cos(a-b) = \frac{1}{2}[\cos(2a) + \cos(2b)].
   \]

2. **Calculate \(\cos(2a)\) and \(\cos(2b)\)** using the double-angle identity:
   
   \[
   \cos(2a) = 2\cos^2 a - 1, \quad \cos(2b) = 2\cos^2 b - 1.
   \]

   Plug in the given values:

   - For \( \cos(2a) \):
     \[
     \cos(2a) = 2\left(\frac{1}{3}\right)^2 - 1 = 2 \times \frac{1}{9} - 1 = \frac{2}{9} - 1 = \frac{-7}{9}.
     \]

   - For \( \cos(2b) \):
     \[
     \cos(2b) = 2\left(\frac{1}{4}\right)^2 - 1 = 2 \times \frac{1}{16} - 1 = \frac{2}{16} - 1 = \frac{-7}{8}.
     \]

3. **Substitute these values** into the original expanded equation:
   
   \[
   \cos(a+b)\cos(a-b) = \frac{1}{2} \left( \frac{-7}{9} + \frac{-7}{8} \right).
   \]

4. **Simplify the result**:

   Find a common denominator for \(\frac{-7}{9}\) and \(\frac{-7}{8}\):
   
   \[
   \frac{-7}{9} + \frac{-7}{8} = \frac{-56}{72} + \frac{-63}{72} = \frac{-119}{72}.
   \]

   Now, substitute this back:

   \[
   \cos(a+b)\cos(a-b) = \frac{1}{2} \times \frac{-119}{72} = \frac{-119}{144}.
   \]

So, the value of \( \cos(a+b)\cos(a-b) \) is:

\[
\boxed{\frac{-119}{144}}.
\]

Would you like further clarification or details on any steps?

### Related Questions:
1. How would the result change if \( \cos a = \frac{1}{2} \) instead of \(\frac{1}{3}\)?
2. What is the product-to-sum identity for sine functions?
3. Can you solve for \(\cos(2a)\) and \(\cos(2b)\) using a different identity?
4. What is the geometrical interpretation of \(\cos(a+b)\)?
5. How would this problem differ if you were given \(\sin a\) and \(\sin b\) instead?

**Tip**: When solving trigonometric expressions, it's useful to memorize key identities such as the product-to-sum and double-angle formulas for simplifying calculations.

Calculate cos(a + b) * cos(a - b) when cos(a) = 1/3 and cos(b) = 1/4.

Learn how to solve a cosine identity problem using the product-to-sum and double-angle formulas. Given cos(a) = 1/3 and cos(b) = 1/4, find the value of cos(a+b) * cos(a-b). This detailed step-by-step guide covers key trigonometric identities and simplifies complex expressions. Suitable for high school students.

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