Gooch, the most prolific run scorer ever in top class cricket, has been used as a batting consultant by England and has also worked extensively back home with Cook, who plays for his former county Essex.
"This 100 is a lot for him, Graham did more stuff back home with me than here. He would think nothing of throwing balls at me for an hour-and-a-half three times a week and we worked hard on changing my technique," Cook told a news conference after scoring his 10th century in 50 Tests.
"I now have slightly different trigger movements, alignment and backlift and they were the three major things to change. It takes a while for it to settle, it's a matter of grooving it."
Cook said England, who hold a first innings lead of 43 runs with five wickets in hand, needed to show similar discipline on Tuesday.
"It was a tough day today and South Africa bowled with a lot of discipline all day. We managed to grind ourselves into a good position, but we must not get ahead of ourselves. We need to get through the first hour tomorrow and then build as big a lead as possible," he said.
Although South Africa's attack tasted success just four times on the third day, coach Mickey Arthur said he was pleased with the bowling effort.
"I thought their efforts today were very commendable and it was very encouraging to see Dale Steyn getting better and better and Jacques Kallis running in and bending his back like he did at the end of the day. England never got away and their scoring rate was always manageable. It was a hard day and I thought we stuck to our task very well," Arthur said.
Arthur added he would be happy if South Africa could bowl England out for less than 500 on Tuesday.
"I'd be very happy with a lead of around 120 to 150. We saw it as an investment session tonight and we hope we get our reward tomorrow morning. If we can manage them to a lead of less than 150 then I'll be very happy," he said.
"When this pitch gets flat it becomes very hard work and all indications are that conditions tomorrow will be the same as today. We musn't let the run-rate get out of hand and hopefully make it a shorter second innings for ourselves."
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