Why did Britain acknowledge the independence of the 13 American colonies?

At first, they didn't. It was war that won political independence for 13 of Britain's North American colonies. After the end of the costly French and Indian War (1763), Britain imposed new taxes and trade restrictions on the colonies, fueling growing resentment and strengthening the colonists' objection to their lack of representation in the British Parliament. Determined to achieve independence, the colonies formed the Continental Army, composed of Minutemen, to challenge Britain's army. The war began in 1775 when Britain sent a force to destroy rebel military stores at Concord, Massachusetts. After fighting broke out, rebel forces began a siege of Boston that ended when Americans forced out the British troops. Britain's offer of pardon in exchange for surrender was refused by the Americans, who declared themselves independent on July 4, 1776. British forces retaliated by driving the American army from New York to New Jersey. On December 25, 1776, The Americans won the battles of Trenton and Princeton. The British army then lost in the Battle of Saratoga in 1777, and in another battle in Monmouth, NJ, in 1778. France (who was already helping America) declared war on Britain in June, 1778. From then on, war continued, and America kept the upper hand. But it wasn't until 1783, in the Treaty of Paris, that Britain acknowledged America's independence, and only because they could not defeat them in war.