(1+i)^{k+1}=2^{\frac{k+1}{2}}{\Bigg \{\cos{\frac{k\pi}{4}\cdot{\cos{\frac{\pi}{4}}}-\sin{\frac{k\pi}{4}}}\cdot{\sin{\frac{\pi}{4}}}+i\Big (\cos{\frac{k\pi}{4}\cdot{\sin{\frac{\pi}{4}}}+\sin{\frac{k\pi}{4}}}\cdot{\cos{\frac{\pi}{4}}}\Big )\Bigg \}