The two have been making such a demand ever since 18 disqualified AIADMK MLAs in September last year met then Governor Ch Vidyasagar Rao and told him that they had lost confidence in Chief Minister Palaniswami.
They had met Rao a day after Palaniswami merged the faction led by him with that of then rebel leader and now Deputy Chief Minister O Panneerselvam.
Dhinakaran, however, sat in the House through the day.
DMK Working President M K Stalin, who is also the Leader of the Opposition, and Dhinakaran, an independent MLA who called himself part of the opposition, reiterated their claim that Palaniswami lacked the numbers to head the government.
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They were apparently referring to the revolt of 18 MLAs who owed allegiance to Dhinakaran and were disqualified by Speaker P Dhanapal last year.
Stalin, who led an opposition-boycott of Purohit's address, claimed it was "clear" that the government had the support of only 111 MLAs in a 235-strong House.
He also expressed his party's displeasure over the governor visiting various districts and holding meetings with state government officials, saying it was "against the Constitution."
Asked if the DMK would move a no-confidence motion against the government, Stalin said he had already made it clear that he would do so if an opportunity arose.
To a question on the possibility of working with Dhinakaran, Stalin said there was "no such necessity" for the DMK.
On the governor's address, he said it had no mention of many issues, including the debt situation and how the government plans to address the decline in industrial growth.
Dhinakaran, who has given a jolt to the ruling AIADMK by defeating its candidate E Madhusudhanan in the December 21 R K Nagar bypoll, claimed that the government had only 111 MLAs on its side and he wants the governor to direct a floor test.
Dhinakaran claimed that the chief minister himself was "not hopeful" of the government fulfilling its term and said the ruling benches "concealed their own fears" by thumping desks in the House today.
He was apparently referring to cheering by AIADMK members whenever Purohit mentioned Palaniswami during his address.
The state government was "afraid" of the Centre, Dhinakaran claimed and described the governor's address as an "eyewash" and a "ritual."
Further, the governor's address did not have mention of the methane extraction project in the Cauvery delta districts, Dhinakaran said, adding the initiative could harm the farmers.
Asked if he would support a no-confidence motion if brought by the opposition, mainly the DMK, against the government, he said that he had already announced extending support to such a move.