In this sentence you should choose "expense" (If you have to travel on company business, we'll pay your expense) because "expense" means the money you will spend. Cost is the price of a specific thing.

expense is a general idea of spending on something, while cost is something might be specific

Since I became a vegetarian my expenses on meat have decreased to 0$ per month. I need to cut down the expenses on video games. Expenses are spent money on something. Also there's another use of this word. "At the expense of smb.", it means somebody pays for you. Yesterday I ate in the restaurant at the expense of my friend. Cost is usually a price of something. Cost of living.

"If you have to travel on company business, we'll pay for your expenses." The difference is that 'cost' refers to the direct amount of money that you have to exchange to get something. For instance, the cost of a Mos Burger is 300yen. But an expense is usually used in business contexts to mean the money you have to pay in order to conduct business. So if you have to pay for a taxi to get to a business conference, the taxi fare is an expense.

how are you? how do you do? what kind of activity do you do? do you go to university or study? what do you do your life? would you like to talk about for you? thanks, have a nice days. ı am an technician at orthodontics department of hacettepe university. ı have been dancing since 1995. salsa, tango, folk dance ... ı like go to the parties three times a week. see you later, good by ;)