～てやる・てみせる: the difference
～てやるN1
Meaning
Just wait and see, Just you wait, To do a favor, For you, Definitely, For sure
Past papers
Example
今夜は俺が奢ってやるから、どんどん好きなもん頼めばいいからな。
I’m going to pay for you, so don’t hesitate and order whatever you want.
てみせるN1
Meaning
I will definitely, I will certainly, Just watch, Whatever it takes, I swear, for sure
Past papers
Tested once in past papers
Example
「何があっても、必ずファンの期待に応えてみせます！」
'No matter what happens, I will definitely live up to my fans' expectations!'
How to tell them apart
「てみせる」: has a feeling of 'doing it for everyone to see, for those who doubt me', full of fighting spirit.
「てやる」: leans to 'doing something for someone', or in a fight a rough 'I'll beat you', as in 「やっつけてやる」 (I'll take you down).