Can a husband naturally practice his marital life and perform tayammum instead of ghusl for a day or two due to cold weather, knowing that he is embarrassed to perform tayammum in front of his colleagues at work?

Intercourse is referenced by activity and ability. A man must ensure his wife's chastity if he fears she might fall into what Allah has forbidden. It is not permissible for one who has had intercourse to suffice with tayammum while water is available, unless he fears significant harm from using cold water, in which case tayammum is permissible for him. If water is not available, the Malikis held that intercourse with one's spouse is not permissible, while the Shafi'is and most scholars permitted it.