The US Has Long Supported Taiwan's Independence...

First the US has always recognized the "One China" policy since the end of WWII. We backed the nationalists who retreated to Formosa (Taiwan) after being swept off the mainland by the communists. We recognized the Nationalist Chinese on Taiwan as the legitimate government of all of China and held out until the Nixon administration in 1971 allowed the PRC to replace the Nationalist Chinese on the UN Security Council. This was part of the Nixon administration's opening up relations with China.

Since then we have supported the Taiwan Chinese militarily and economically while trying to maintain a schizoid policy of "One China" but not advocating either the Nationalist Chinese on Taiwan taking over all of China, or the PRC taking over Taiwan. (by the way Taiwan was never part of China until occupied by the Nationalists when they fled the mainland).

Taiwan sticks in the craw of the PRC because they have been so successful economically. That example is seen as a direct threat to the totalitarian PRC.

The US had no formal agreement for the defense of Taiwan yet we have always strongly supported them and left the impression that we would come to their aid militarily. We (the USA) have been trying to keep a foot in both camps without formally committing to anything. We have no formal defense agreement with Taiwan, yet we leave the impression that we will defend it in the hopes that the threat will be enough. It seems that we are nearing a time when the PRC wants to call our bluff.

When the US was overwhelmingly dominant militarily and economically we could get away with it. The fear that we might step in was enough to deter PRC aggression against Taiwan. (and the whole region) The waffling appearance of weakness on the part of our leadership increases the chances of a miscalculation that will lead to war.