odpoveď:
Pozri nižšie.
vysvetlenie:
nechať # 1 + costheta + isintheta = r (cosalpha + isinalpha) #, tu # R = sqrt ((1 + costheta) ^ 2 + sin ^ 2Theta) = sqrt (2 + 2costheta) #
= #sqrt (2 + 4cos ^ 2 (theta / 2) -2) = 2cos (theta / 2) #
a # Tanalpha = sintheta / (1 + costheta) == (2sin (theta / 2) cos (theta / 2)) / (2cos ^ 2 (theta / 2)) = tan (theta / 2) # alebo # Alfa = theta / 2 #
potom # 1 + costheta-isintheta = r (cos (-a) + ISIN (-a)) = R (cosalpha-isinalpha) #
a môžeme písať # (1 + costheta + isintheta) ^ n + (1 + costheta-isintheta) ^ n # pomocou vety DE MOivre ako
# R ^ n (cosnalpha + isinnalpha + cosnalpha-isinnalpha) #
= # 2r ^ ncosnalpha #
= # 2 * 2 ^ NCOs ^ n (theta / 2) cos ((ntheta) / 2) #
= # 2 ^ (n + 1) cos ^ n (theta / 2) cos ((ntheta) / 2) #
