てみせる

1. 1Doing an action in front of someone as a demonstration, showing them how it is done.
2. 2Declaring firm resolve to accomplish something and prove it to others or to oneself, often with a sense of 'you'll see.'Ex. 1

Neutral in register. The determination sense usually appears in the speaker's own declarations, often with 必ず, 絶対に or 今度こそ, and is typical of promises, pep talks and vows before a challenge or after an earlier failure. The demonstration sense is used when literally showing someone how something is done. Unlike てみる, which means only trying something out, てみせる stresses proving or showing the result.

• てみるN4「てみる」 means 'try doing and see,' with no certainty about the result, focusing on the attempt or experience; 「てみせる」 means doing something to show people or declaring you will definitely do it, focusing on convincing others. They share the same conjugation and neutral register, but are not interchangeable.
• てやるN4When expressing determination, the two are often interchangeable, and both attach to the て-form of a verb. 「てみせる」 is neutral in register and focuses on proving something to others; 「てやる」 is colloquial and often carries anger or defiance. 「てやる」 can also express doing a favor or deliberately displeasing someone.
• ないではおかない / ずにはおかないN1「ずにはおかない」 also expresses determination, but with a strong, somewhat written tone, attaching to the ない-form minus ない; 「てみせる」 attaches to the て-form, is neutral in register, and carries the nuance of proving something to others. The former also means 'inevitably make someone...' — a meaning the latter lacks.
• ようと思うN4「ようと思う」 attaches to the volitional form and simply states one's own intention in a mild tone; the determination in 「てみせる」 is stronger and more outwardly expressed, with a nuance of declaring or proving to others. 「ようと思っている」 can also describe a third person's intention.
• んばかりにN1「んばかりに」 attaches to the negative stem, is written-style, and often exaggerates a look or manner 'as if about to...' without the action actually occurring; 「てみせる」 attaches to the て-form and means actually doing something to show others or the speaker's determination, so they cannot be interchanged.
• ものかN2「ものか」 also expresses strong will, but as negation: 'absolutely not...,' and can also mean 'how could...'; 「てみせる」 is positive determination, 'I will definitely do it.' 「ものか」 attaches to the dictionary form, た-form, and so on, and leans colloquial; they cannot be interchanged.

てみる only means trying something out and ようと思う states a plain intention, whereas てみせる vows to achieve a result and prove it. Defiant てやる can also carry an 'I'll show them' nuance but sounds rougher and more aggressive; ないではおかない / ずにはおかない is formal and can express either inevitability or firm resolve (真相を突き止めずにはおかない); ものか is a negative vow ('no way I'll…'); and んばかりに means 'as if about to,' describing appearance rather than making a vow.