Evidence to show? or just that clause?

Hello,

I have a question about what words follow this noun 'evidence'. In the following sentence, "There is evidence to show that she entertained the idea of suicide long before this", 'to show' follows 'evidence' and I have seen a lot of cases without 'to show', like that clause follows 'evidence' without 'to show': ex) Do you have evidence that your husband is cheating on you?

I'm wondering if it's okay to use the first sentence without 'to show' and it still means the same.

Also, I'm wondering too why to infinitive was used in that clause. I think that 'showing' would be more appropriate. Because showing would mean 'which shows'.

Thanks so much, I would really appreciate any help.