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.