<![CDATA[Purpose: This study aims to analyze students’ learning trajectories in constructing proofs of trigonometric identities through the lens of didactic transposition. Method: This research employed a descriptive qualitative design involving undergraduate students enrolled in a mathematics education program who were taking a trigonometry course. Participants were selected purposively based on their active engagement in proof-related learning activities. Data were collected through classroom observations, analysis of students’ written proof documents, and in-depth interviews to capture students’ reasoning processes and conceptual interpretations. The collected data were analyzed through qualitative procedures consisting of data reduction, data display, and conclusion drawing to identify patterns in students’ learning trajectories during the proof construction process. Findings: The findings indicate that students’ learning trajectories in constructing trigonometric identity proofs develop through three interconnected phases. The first phase involves exploring the meaning of trigonometric ratios within geometric contexts. The second phase focuses on formalizing the relationships among trigonometric ratios and identities. The third phase involves constructing deductive arguments to establish the fundamental identity , reflecting a shift from empirical reasoning to formal mathematical proof. Significance: The didactic transposition process plays a crucial role in transforming students’ understanding from procedural manipulation toward deeper conceptual meaning. This process facilitates the emergence of structured learning trajectories that support reflective reasoning and higher-order mathematical thinking. The findings suggest that trigonometry instruction should emphasize conceptual learning trajectories grounded in didactic transposition to strengthen students’ proof construction abilities and promote meaningful mathematical understanding.]]>
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