If cos α = 1/2(a + 1/a), cos β = 1/2(b + 1/b), find the value of cos(α - β).

Step 1: Write down the given expressions for $\cos \alpha$ and $\cos \beta$.
We are given:

$$\cos \alpha = \frac{1}{2}\left(a + \frac{1}{a}\right)$$ $$\cos \beta = \frac{1}{2}\left(b + \frac{1}{b}\right)$$

Step 2: Find expressions for $\sin \alpha$ and $\sin \beta$.
We use the identity $\sin^2 \theta = 1 - \cos^2 \theta$.
First, calculate $\cos^2 \alpha$:

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