Correct Option - 3

Here, the action of 'insisting' requires an appropriate preposition following it. 

Option 1 is incorrect because we use 'with' when we want to ASSOCIATE TWO things / PEOPLE with each other. Here, there is no such indication.

Option 2 is incorrect because the word 'from' is used to indicate origin (Eg: 'She is from PUNE'), a time period (Eg: 'The classes will be held from Monday to Friday') or in the format 'From-to' (Eg: 'She travelled from Mexico to Canada'). Since there is no indication of a similar context in the given SENTENCE, 'from' is incorrect. 

Option 4 is incorrect because we generally use 'at' when we want to specify a time or location. Here, the word 'evidence' is a noun which does not indicate location. We would use 'at' if a location had been specified, such as: 'The detective insisted on checking the evidence at the police station'. Here, the noun 'evidence' is located at the police station and the preposition 'at' describes the same.

Option 3 is correct because the preposition 'on' is used whenever we use the verb 'insist'.

Hence the correct sentence is, 'The detective insisted on checking the evidence thoroughly'.