Step-by-step solution:

1. Analyze the definition of a wavefront. A wavefront is defined as the locus of all points in a medium that have the same phase of oscillation. In other words, for any two points on the same wavefront, their phases are equal.

2. Apply the definition to the given wavefronts.

   • The problem states that the line cd represents a wavefront of the incident light wave in medium-1. The points c and d both lie on this wavefront. Therefore, their phases must be equal. $$$$

entail \phi_{c} = \phi_{d} \quad \cdots(1) entail $- Similarly, the line `ef` represents a wavefront of the refracted light wave in medium-2. The points `e` and `f` both lie on this wavefront. Therefore, their phases must also be equal.$ entail \phi_{e} = \phi_{f} \quad \cdots(2) entail $$

3. Evaluate each option based on these facts.

   • Option A: $\phi_{c}$ cannot be equal to $\phi_{d}$ This statement is incorrect. As established in equation (1), since c and d are on the same wavefront, their phases must be equal.

   • Option B: $\phi_{d}$ can be equal to $\phi_{c}$ This statement is true. In fact, $\phi_{d}$ must be equal to $\phi_{c}$. While technically correct, it's a weak statement and might not be the best description of the physics involved compared to other options.

   • Option C: $(\phi_{d}-\phi_{f})$ is equal to $(\phi_{c}-\phi_{e})$ Let's test this statement using our established equalities. We can rearrange the equation to be tested:

     $$$$

entail \phi_{d} - \phi_{f} = \phi_{c} - \phi_{e} entail We can substitute $\phi_{d} = \phi_{c}$ (from eq. 1) and $\phi_{f} = \phi_{e}$ (from eq. 2) into the left-hand side (LHS) of the equation: entail \text{LHS} = \phi_{d} - \phi_{f} = \phi_{c} - \phi_{e} entail $$ The right-hand side (RHS) is $\phi_{c} - \phi_{e}$. Since LHS = RHS, the statement is a mathematical identity based on the properties of wavefronts. Thus, this option is correct.

    *Alternative physical argument for C:*    According to Huygens' principle, the time taken for light to travel from one wavefront to a subsequent one is the same along any ray. The rays are perpendicular to the wavefronts. So, the time taken to travel from point `c` on wavefront `cd` to point `e` on wavefront `ef` is the same as the time taken to travel from point `d` to point `f`.    $$ 

entail t_{c \rightarrow e} = t_{d \rightarrow f} entail The phase difference $\Delta\phi$ is related to the time interval $\Delta t$ by $\Delta\phi = \omega \Delta t$, where $\omega$ is the angular frequency (which remains constant during refraction). Therefore, the phase change from `c` to `e` is equal to the phase change from `d` to `f`. entail \phi_{e} - \phi_{c} = \phi_{f} - \phi_{d} entail $Rearranging this equation gives:$ entail \phi_{d} - \phi_{f} = \phi_{c} - \phi_{e} entail $$ This confirms that statement C is correct based on the principles of wave propagation.

*   **Option D: $(\phi_{d}-\phi_{c})$ is not equal to $(\phi_{f}-\phi_{e})$**    Let's evaluate the terms in the parentheses using our initial findings:    -   $\phi_{d} - \phi_{c} = \phi_{c} - \phi_{c} = 0$    -   $\phi_{f} - \phi_{e} = \phi_{e} - \phi_{e} = 0$    The statement claims that `0` is not equal to `0`, which is false.

4. Conclusion Both options B and C are technically correct statements. However, in a single-choice question, we must select the best description. Option B is a simple consequence of the definition of a single wavefront. Option C is a more comprehensive statement that relates the phases of all four points across both media, describing a property of the propagation and refraction process itself. Therefore, C is the better and intended answer. The given condition $\phi_{c} \ne \phi_{f}$ just confirms that cd and ef are distinct wavefronts separated by a finite propagation time, as expected.