2.5.3—de Moivre's theorem

Expand (cosθ+i sinθ)^n using the binomial theorem, then equate its real and imaginary parts with cos nθ+i sin nθ to obtain trigonometric identities.

Separate even powers of i for the real part and odd powers for the imaginary part. Keep the combinatorial coefficients and signs organised before simplifying.

For n=2, the real part of (cosθ+i sinθ)^2 gives cos2θ=cos²θ−sin²θ, while the imaginary part gives sin2θ=2sinθcosθ.

The identity follows from equality of complex numbers, not from treating i as an ordinary positive number.