plot explanation - Why did Jose give Cole the gun? - Movies & TV

At the airport Cole meets Jose who gives him a gun and instructs him to kill Dr. Peters. Now the question is - why?

They make it clear in the film that the past cannot be altered and that is why he was sent to get a sample of the virus and not stop it.

Answer

The scientists use Jose to force Cole into trying to kill the virus-spreader because they needed to learn the virus-spreader's identity.

Remember, the scientists knew Cole was at the airport because Cole had left a voicemail from there. But in the voicemail, while Cole said the Army of the 12 Monkeys did not spread the virus, he didn't say who the real culprit was.

So Jose's task at airport is to give Cole the gun and then watch who Cole goes after. But while Jose and Cole are talking, Catherine runs up and says she's figured out it is the redhead assistant to Dr Goynes who spreads the virus. At this point, Jose fades out of the scene, having got what he needed. (for confirmation see scene as written in production draft of script)

Of course, airport security shoots Cole when Cole points the gun at the virus-spreader. The scientists may have provided Cole a non-working gun to encourage this outcome.