Proving Fermat’s Last Theorem via Complex Numbers and Trigonometry

We present a geometric and analytic reformulation of Fermat’s Last Theorem (FLT) using complex numbers and trigonometric identities. Starting from the normalized form (a/c) ⁿ + (b/c) ⁿ = 1, we define a complex number z = (a/c) ^ (n/2) + i (b/c) ^ (n/2) of unit modulus. This construction implies z = eiθ, leading to a pair of constraints: a modulus identity and a tangent identity tan (θ) = (b/a) ^ (n/2). We demonstrate that these constraints cannot be satisfied simultaneously when n 2, due to conflict between algebraic and transcendental values. This contradiction offers a simple and intuitive route to the nonexistence of nontrivial integer solutions, providing an accessible geometric perspective on FLT—possibly aligned with Fermat’s original intuition.