Question

(cosα+isinα)^n+(cosα-isinα)^n=2cosnα

Answer 1

运用公式e^iα=cosα+isinα

由这个式子可以得到：

e^(-iα)=cos(-α)+isin(-iα)=cosα-isinα

对原式进行化简,得

(cosα+isinα)^n+(cosα-isinα)^n

=(e^iα)^n+[e^(-iα)]^n

=e^(inα)+e^(-inα)

=cos(nα)+isin(nα)+cos(-nα)+isin(-nα)

=cos(nα)+isin(nα)+cos(nα)-isin(nα)

=2cos(nα)