Which of these events did not occur during the presidency of william howard taft?

Since you haven't given the choices in your assigned question, the best I can do here is offer you a list of some of the key events that DID occur during the years of William Howard Taft's presidency.

Robert Peary led an expedition that was the first to reach the North Pole (in April, 1909).

Nellie Taft (the president's wife) suffered a stroke in May, 1909.  She recovered but it took many months.

In July, 1909, Congress passed the proposal for the 16th Amendment and submitted it to state legislatures for ratification. The amendment allowed Congress to collect income taxes.  Ratification was complete by 1913, while Taft was still in office.

In May, 1911, the Supreme Court ordered the dissolution of the Standard Oil Company for violating the Sherman Antitrust Act.  The company was required to split into smaller corporations.   The American Tobacco Company was similarly ordered to dissolve for antitrust violations later in that same month.

In January, 1912, New Mexico became the 47th state of the United States, and in February, 1912, Arizona became the 48th state.

The famous sinking of the Titanic occurred in April, 1912.

There were other events that could be mentioned, but that gives you some ideas of the history of the time.

The answer to that question is a