2024 SJI P1 Q7

By considering the expansion of \((1+\mathrm i)^{2n}\), where \(\mathrm i^2=-1\), show that

\[{}^{2n}\mathrm C_0-{}^{2n}\mathrm C_2+{}^{2n}\mathrm C_4-{}^{2n}\mathrm C_6+\cdots+(-1)^n{}^{2n}\mathrm C_{2n}=2^n\cos\left(\frac{n\pi}2\right)\quad\text{for all}\hspace{0.5em}n\in\mathbb Z^+.\][7]