Using complex numbers, derive the trigonometric identities for cos(2θ) and sin(2θ).

When dealing with complex numbers and trigonometric functions, always turn to DeMoivre's Theorem that states [cos(θ)+isin(θ)]ⁿ = [cos(nθ)+isin(nθ)]. If we set n=2, the we see a combination of cos(2θ) and sin(2θ) on the right hand side. From here, we can expand the left hand side, just like we would with a normal quadratic expression, giving us: cos²(θ) + 2cos(θ)(isin(θ)) + (isin(θ))². This can then be simplified to cos²(θ) - sin²(θ) + 2cos(θ)(isin(θ)) as i² = -1 by definition. Combining the right hand side and the left hand side gives: cos²(θ) - sin²(θ) + 2cos(θ)(isin(θ)) = cos(2θ)+isin(2θ) We can then equate real and imaginary parts of the equations to give: cos²(θ) - sin²(θ) = cos(2θ) and 2cos(θ)(isin(θ)) = isin(2θ), and therefore 2cos(θ)sin(θ) = sin(2θ).