A Matrix-Based Proof of a Trigonometric Identity via the Cayley Transform on SO(2)

We present a matrix-based proof of a classical trigonometric identity by interpretingcos x and sin x as entries of planar rotations in SO(2). The key tool is the Cayley transform,which maps skew-symmetric matrices to special orthogonal matrices and yields a rationalparametrization of planar rotations (excluding the π-rotation for finite parameter values).This parametrization, equivalent to the tangent half-angle (Weierstrass) substitution, reducesboth sides of the identity to the same rational expression t2. For reproducibility, we reportan independent symbolic verification in SymPy, where the difference between both sidessimplifies to zero.