"Though" is used to contrast two ideas. The formula is: [Though]+[Idea 1], [Idea 2]. "Though he is able to provide for his family, he does work too hard." "Though he studied hard for the test, he still only got a 'B' grade." By 'trough" I think you mean "through." It is a preposition that means "into one side or end of an entrance, passage, hole etc and out of the other side or end." "William Tell shot an arrow through the apple." There are some other uses too, which are available here: http://www.ldoceonline.com/dictionary/through_1 If you'd ever like a lesson on giving reasons, using prepositions, or any other grammar point, let me know and I would be glad to work with you. :)

through = from one end to another trough =long narrow container for animal to drink

Though it is not easy, through hard work we will succeed!!! They ran through the town in the dark. Though they has done nothing wrong, the police chased them. All through the night they felt fear, though they were innocent.

I agree with the other answers, yet there is also an expression "through and through", which is used to mean "completely", "thoroughly" or "in all possible ways." Eg. "I know him through and through - he'd never do something like that." "She's a Millwall supporter through and through." "Host City's Success, Through and Through" (headline of a Chicago Tribune article on the London Olympics, just posted yesterday)

Though, pronounced like « ذوه ». "Though" can usually be used the same as "although". Example: Though it was early in the morning, he craved a slice of pizza. (In Jim Roger's post above, he gives the example, "Though he studied hard for the test, he still only got a 'B' grade", which is correct. This can also be written as, "He only got a 'B' grade, though he studied hard for the test". So, the formula Jim Rogers gives is correct, but not the only way to use "though".) Through, pronounced like « ثْروه ». "Through" means, "from one end to another". Example: The bullet went through his heart. Trough, pronounced like « تْرُف ». A "trough" is a long bucket or container with an open top, often used for feeding farm animals. Example: The politicians were like pigs at a feeding trough, feeding on the public's money. Pigs at a feeding trough: http://img21.imageshack.us/img21/1174/superstock25530816.jpg