神马品牌网上海儿童好玩的地方:用数学归纳法证明:cos(x/2)×cos(x/2^2)×...×cos(x/2^n)=sinx/[2^n×sin(x/2^n)]
来源：百度文库 编辑：神马品牌网 时间：2024/05/09 21:21:04
当n=k+1时,怎么证的??
设当n=k时成立，则有cos(x/2)×cos(x/2^2)×...×cos(x/2^k)=sinx/[2^k×sin(x/2^k)]
则当n=k+1时，cos(x/2)×cos(x/2^2)×...×cos(x/2^k)×cos[x/2^(k+1)]=sinx×cos[x/2^(k+1)]/[2^k×sin(x/2^k)]
将sin(x/2^k)＝2×cos[x/2^(k+1)]×sin[x/2^(k+1)]代入式子消掉cos[x/2^(k+1)]得
cos(x/2)×cos(x/2^2)×...×cos(x/2^k)×cos[x/2^(k+1)]＝sinx/[2^(k+1)×sin(x/2^(k+1))]
即当n=k+1时等式也成立