(1) 对f(x) = cos(x)+isin(x),成立f(x+y) = f(x)f(y).
f(x)f(y) = (cos(x)+isin(x))(cos(y)+isin(y))
= (cos(x)cos(y)-sin(x)sin(y))+i(cos(x)sin(y)+sin(x)cos(y))
= cos(x+y)+isin(x+y)
= f(x+y).
(2) (f(x))^n = f(nx).
证明使用数学归纳法.
首先对n = 1显然成立.
假设对n = k成立,即有(f(x))^k = f(kx).
则(f(x))^(k+1) = f(x)·(f(x))^k = f(x)f(kx) = f((k+1)x),即得n = k+1时成立.
因此结论对任意正整数k成立.
(3) 所求式 = f(π/12)f(5π/12)+(f(π/6))^2007
= f(π/12+5π/12)+f(2007π/6)
= f(π/2)+f(669π/2)
= cos(π/2)+isin(π/2)+cos(669π/2)+isin(669π/2)
= 2i.