To find the value of \( \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} \) given \( \tan \theta = \frac{8}{15} \), we need to determine \(\sin \theta\) and \(\cos \theta\). Using the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), and given \(\tan \theta = \frac{8}{15}\), we represent \(\sin \theta = 8k\) and \(\cos \theta = 15k\) for some \(k\).

We know the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\).

Substituting the values, we have:

\((8k)^2 + (15k)^2 = 1\)

\(64k^2 + 225k^2 = 1\)

\(289k^2 = 1\)

\(k^2 = \frac{1}{289}\)

\(k = \frac{1}{17}\)

This gives \(\sin \theta = 8 \times \frac{1}{17} = \frac{8}{17}\) and \(\cos \theta = 15 \times \frac{1}{17} = \frac{15}{17}\).

Now, calculate \( \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} \):

\( \sqrt{\frac{1 - \frac{8}{17}}{1 + \frac{8}{17}}} = \sqrt{\frac{\frac{17-8}{17}}{\frac{17+8}{17}}} = \sqrt{\frac{9}{25}} = \frac{3}{5}\).

Thus, the value is \(\frac{3}{5}\).