He . . . . . . . in wearing the old fashioned coat inspite of his wife’s disapproval.

The correct answer is C. persists.

• Resists means to refuse to do something, especially something that is difficult or unpleasant.
• Desists means to stop doing something, especially something that is wrong or harmful.
• Persists means to continue doing something even though it is difficult or unpleasant.
• Insists means to demand or request something strongly.

In this case, the man is continuing to wear the old-fashioned coat even though his wife disapproves. This suggests that he is persisting in his behavior, despite the fact that it is causing him some difficulty.