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I think there seems to be a difference though it'd be very little.
1.Do you think so?
In my opinion, it means similar to "is this your way of thinking?"
2.Do you think like that(this)?
Somewhat differently, I guess it means similar to "is that(this) like your thought?"
And I want to know how to use them correctly in answer to a question.
Nowadays, so in that usage requires as antecedent an assertion or a set of related assertions.
He's not answering the phone. He's probably in the shower.
--Do you think so? He takes that phone everywhere.
Or
In the November elections we're going to see college-educated suburban women turn out in large numbers, but also an increased turnout by white males without college education.
-- You think so?
Whereas "like that" can refer to a more general pattern of thought.
I don't know why she's not answering the phone. Maybe the house burnt down, or she's been kidnapped, or maybe she been in a car accident.
--There's no reason to think like that. Her phone just might be out of power.
I've spoken with non-native English speakers who have used both of those phrases interchangeably. I would agree with your analysis that they are slightly different.
Colloquially, do you think so would be an inquisition to another person's opinion to confirm their belief.
Do you think like that is a more cumbersome sentence structure, and less conversational. I would take it as an implication that the person asking the question was trying to find out if my thoughts follow a certain pattern or methodology, as opposed to simply confirming my opinion.
