pronoun + particle (instrumental singular of ta + emphatic/affirmative particle)
Analysis. tāya (instrumental singular of the pronoun ta, 'that') + api (emphatic particle 'even, also, and'). The combination emphasizes the instrumental relationship: 'by that very thing', 'even by that'.
Etymology. tāya is the instrumental form of ta (demonstrative pronoun cognate with Sanskrit tat, Greek to-, Latin is-te 'that'; from Proto-Indo-European *to-d). Api is a particle of affirmation and emphasis, unrelated to the pronoun. Together they form a strengthened instrumental expressing cause or means with emphasis on the particular agent or instrument involved.
Meanings from the Digital Pāḷi Dictionary; morphology and etymology by Claude.
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