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In Tamil language, there is a proverb for a particular sequence of actions performed. The proverb is, "Pillaiya killi vittu, thottila aatradhu", meaning, "Pinching a child and then oscillating the child's hammock". (Rough translation)

This is usually said when politicians instigate something controversial and then they themselves try to pacify the situation. People who are innocent, wouldn't even know that they are being manipulated by the politicians.

Is there any equivalent saying in English? The closest I could find is "Tiger in a sheep's clothing". But I am not really convinced with it since I think it doesn't exactly capture this behaviour.

The core idea in your proverb seems to be similar to that of the "arsonist firefighter." Such a person is said to be

setting a fire in order to put it out

An example of this expression used in the context of the phenomenon it describes appears in Andrew Murr, "A Moth to the Flame," in Newsweek (June 30, 2002):

Sadly, the scenario is not that rare; in the last year alone, more than a half-dozen firefighters around the country have been charged with starting a blaze, or intending to. "The big one is the vanity hero type," says Doug Allen, an arson expert—the firefighter who starts blazes just so he can put them out.

A figurative use of a similar phrase appears in Helene Stapinski, "'Transmission' Shifts Between Clumsy and Charming," in the Chicago Tribune [June 24, 2004):

Like a firefighter who sets a blaze in order to put out the flames and appear heroic, Arjun [Mehta, the novel's ptotagonist] unleashes a computer virus in the hopes that he will solve the problem and be rehired.

I don't think that the expression "setting a fire in order to put it out" has yet attained the status of a modern proverbial phrase in English, but I do think that it may be well on its way toward such status.

I cannot think of any similar idiom that uses behaviours (e.g., pinching, rocking) to convey the idea that seems to be conveyed by the Tamil saying. However, Janus-faced, and equivalently but less poetically, two-faced, both seem to convey the idea of duplicitous, contradictory behaviour.

I'd paraphrase what your politician is doing as "creating a problem so that he can solve it". It's not an idiom though. So I googled it. And having typed "create a problem", google suggested an autocomplete:

create a problem sell a solution

Google reckons it has 290 million hits... but most of them are using words from the phrase not the phrase itself. By using quote marks to restrict it to sites including the words as two separate phrases, there are only 320 hits.

So it's not a widely used phrase, so you can't really call it an idiom. It's catchy though, and clear what it means. I might use it, or something similar, maybe:

Create a problem so that you can offer the solution