Which American TV show featured the antics of identical cousins played by the same actor?

"The Patty Duke Show" is an American television sitcom which ran on ABC from September 18, 1963, to April 27, 1966.

The series was developed as a vehicle for teenage star Patty Duke, (December 14, 1946 – March 29, 2016). Duke starred in dual roles of "twin cousins" Patty and Cathy Lane.

The network enlisted writer Sidney Sheldon to create the show. He had Duke spend a week with his family at their home to generate ideas. He noticed she had two distinct sides to her personality (later in life she would be diagnosed as manic-depressive), and came up with the identical cousins idea.

Patty Lane (Duke) is a normal, chatty, rambunctious teenager who lives in the Brooklyn Heights section of New York City. Her "identical paternal cousin," Cathy Lane (also played by Duke), is sophisticated, brainy and demure.

While the girls are physically identical, their style, tastes and attitudes are nearly opposite, leading to the comedic situations. The remarkable physical resemblance that Patty and Cathy share is explained by the fact that their fathers are identical twins. While Patty speaks with a typical American accent, Cathy speaks with a general European accent.

In development, Cathy was initially from Scotland. So Duke learned a true Scottish burr for the Cathy character. However, the accent was done "so well they couldn't understand me".

According to Duke, the show successfully captured her personality in the two characters.