〜てみせる
show (how); I'll definitely (you'll see)
Verb て-form + みせる
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てみせる has two uses, both built on 見せる, "to show".
Demonstrate: doing something so others can see how: 先生が手本を書いてみせた, "the teacher wrote it out to show us how". やってみせてください means "please show me how".
Determination: declaring you'll achieve something, as if to prove it to others: 今度こそ、絶対に合格してみせる, "this time I'll pass, you'll see". This use is common in anime and dramas.
It can also mean putting on an appearance: 笑ってみせた, "put on a smile".
Don't confuse it with てみる, "try and see". みせる is about other people watching.
Examples
今度こそ、絶対に合格してみせる。
This time, I'll pass for sure. You'll see.
先生が手本を書いてみせた。
The teacher wrote it out to show us how.
必ず勝ってみせます。
I'll win, I promise you.
彼は笑ってみせたが、本当は悲しかった。
He put on a smile, but he was really sad.
使い方をやってみせてください。
Please show me how to use it.