In the process of sodium fusion, organic compounds are fused with sodium metal to convert them into ionic form. The fusion extract is then boiled with concentrated nitric acid for testing halogens, and the purpose of boiling with nitric acid is to achieve several things:
1. Decomposing Na\(_2\)S and NaCN: When sodium sulfide (Na\(_2\)S) or sodium cyanide (NaCN) are formed during the fusion, boiling with concentrated nitric acid will decompose them. This is important because these compounds might interfere with the halogen detection process, so they must be decomposed to avoid any false results.
2. Increasing solubility of AgCl: While nitric acid might increase the solubility of certain compounds, it is not primarily responsible for increasing the solubility of AgCl. The solubility of AgCl is more directly affected by the presence of other chemical reagents.
3. Increases concentration of NO\(_3^{-}\) ions: Boiling with concentrated nitric acid does increase the concentration of NO\(_3^{-}\) ions, but this is not the main purpose for boiling when testing halogens, which focuses more on decomposition of interference compounds.
4. Helps in precipitation of AgCl: The formation of AgCl is typically achieved by adding silver nitrate to the solution, but the boiling step with nitric acid is not primarily aimed at precipitating AgCl.
Thus, the correct answer is:
\[
\text{(3) decomposes Na}_2\text{S and NaCN, if formed.}
\]