Step 1: Addition of HCl to the alkene.
The given alkene is \(1\)-methylcyclohexene.
On reaction with HCl, Markovnikov addition takes place.
Hydrogen adds to the carbon having more hydrogens, while chlorine attaches to the more substituted carbon.
Thus, the product formed is:
\[
1\text{-chloro-1-methylcyclohexane}
\]
This is a tertiary alkyl halide.
Step 2: Reaction with \(AgNO_2\).
Silver nitrite is a covalent reagent and generally forms alkyl nitrites:
\[
R-ONO
\]
instead of nitro compounds:
\[
R-NO_2
\]
Thus,
\[
1\text{-chloro-1-methylcyclohexane}
\]
reacts with
\[
AgNO_2
\]
to form the corresponding alkyl nitrite:
\[
R-ONO
\]
Step 3: Identify the correct product.
The product contains:
\[
-ONO
\]
attached to the tertiary carbon bearing the methyl group.
This corresponds to option (4).
Step 4: Final conclusion.
Hence, the major product formed is the alkyl nitrite:
\[
\boxed{\text{Option (4)}}
\]