If `cos(alpha+beta)=(4)/(5) and sin(alpha-beta)=(5)/(13)` , where `alpha` lie between 0 and ` (pi)/(4)`, then find that value of `tan2alpha`.

Explanation

To find the value of \( \tan(2\alpha) \) given \( \cos(\alpha + \beta) = \frac{4}{5} \) and \( \sin(\alpha - \beta) = \frac{5}{13} \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
From the given \( \sin(\alpha - \beta) = \frac{5}{13} \), we can find \( \cos(\alpha - \beta) \) using the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \).

\[ \cos^2(\alpha - \beta) = 1 - \sin^2(\alpha - \beta) = 1 - \left(\frac{5}{13}\right)^2 \]
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