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To solve the given problem, we need to find the value of \(\cos(\alpha - \beta)\) given that \(\cos(\alpha + \beta) = m\) and \(\tan \alpha \tan \beta = 2\).

First, let's recall the sum and difference formulas for cosine:

\[
\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta
\]
\[
\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta
\]

Given \(\cos(\alpha + \beta) = m\), we can write:
\[
m = \cos \alpha \cos \beta - \sin \alpha \sin \beta
\]

From \(\tan \alpha \tan \beta = 2\), we know:
\[
\tan \alpha = \frac{\sin \alpha}{\cos \alpha}, \quad \tan \beta = \frac{\sin \beta}{\cos \beta}
\]
\[
\tan \alpha \tan \beta = \frac{\sin \alpha \sin \beta}{\cos \alpha \cos \beta} = 2 \implies \sin \alpha \sin \beta = 2 \cos \alpha \cos \beta
\]

Next, substitute this relationship into the difference formula:
\[
\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta
\]
\[
\cos(\alpha - \beta) = \cos \alpha \cos \beta + 2 \cos \alpha \cos \beta
\]
\[
\cos(\alpha - \beta) = 3 \cos \alpha \cos \beta
\]

Since \(\cos(\alpha + \beta) = m\), we have:
\[
m = \cos \alpha \cos \beta - 2 \cos \alpha \cos \beta = -\cos \alpha \cos \beta
\]
\[
\cos \alpha \cos \beta = -\frac{m}{1}
\]

Finally, substituting \(\cos \alpha \cos \beta = -\frac{m}{1}\) into the expression for \(\cos(\alpha - \beta)\):
\[
\cos(\alpha - \beta) = 3 \left( -\frac{m}{1} \right)
\]
\[
\cos(\alpha - \beta) = -3m
\]

So, the correct answer is:
\[
\boxed{-3m}
\]

Would you like any further details or have any questions?

Here are some other questions you might be interested in:
1. How do you derive the sum and difference formulas for trigonometric functions?
2. What are some applications of trigonometric identities in real-world problems?
3. Can you solve an example problem involving the double-angle formulas?
4. How do you convert between degrees and radians in trigonometry?
5. What is the relationship between the trigonometric functions and the unit circle?

**Tip:** When solving trigonometric problems, it is helpful to remember and utilize fundamental identities and formulas. This can simplify the process significantly.

Learn how to solve trigonometric equations using sum and difference formulas. This problem involves finding \(\cos(\alpha - \beta)\) given \(\cos(\alpha + \beta) = m\) and \(\tan \alpha \tan \beta = 2\). Explore detailed step-by-step solutions and applications of trigonometric identities. Suitable for advanced high school students.

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