(a) CoCl$$_3$$ · 4NH$$_3$$,
(b) CoCl$$_3$$ · 5NH$$_3$$,
(c) CoCl$$_3$$ · 6NH$$_3$$ and
(d) CoCl(NO$$_3$$)$$_2$$ · 5NH$$_3$$
Number of complex(es) which will exist in cis-trans form is/are

We need to identify the coordination compounds that can exhibit cis-trans (geometrical) isomerism.

Compound (a): CoCl$$_3$$ · 4NH$$_3$$

Using Werner's theory, the complex is $$[\text{Co}(\text{NH}_3)_4\text{Cl}_2]\text{Cl}$$.

This is of the type $$[\text{MA}_4\text{B}_2]$$ in an octahedral geometry. In this arrangement, the two Cl ligands can be adjacent (cis) or opposite (trans).

Therefore, this compound exhibits cis-trans isomerism.

Compound (b): CoCl$$_3$$ · 5NH$$_3$$

The complex is $$[\text{Co}(\text{NH}_3)_5\text{Cl}]\text{Cl}_2$$.

This is of the type $$[\text{MA}_5\text{B}]$$, which has only one possible arrangement in octahedral geometry. No cis-trans isomerism is possible.

Compound (c): CoCl$$_3$$ · 6NH$$_3$$

The complex is $$[\text{Co}(\text{NH}_3)_6]\text{Cl}_3$$.

This is of the type $$[\text{MA}_6]$$, with all identical ligands. No geometrical isomerism is possible.

Compound (d): CoCl(NO$$_3$$)$$_2$$ · 5NH$$_3$$

The complex is $$[\text{Co}(\text{NH}_3)_5(\text{NO}_3)](\text{NO}_3)\text{Cl}$$.

This is of the type $$[\text{MA}_5\text{B}]$$, which does not show cis-trans isomerism.

Only compound (a) exhibits cis-trans isomerism.

Hence, the number of complexes existing in cis-trans form = $$\boxed{1}$$