求cosπ/9*cos2π/9*cos3π/9*cos4π/9

求cosπ/9*cos2π/9*cos3π/9*cos4π/9
求cosπ/9*cos2π/9*cos3π/9*cos4π/9

求cosπ/9*cos2π/9*cos3π/9*cos4π/9
cosπ/9*cos2π/9*cos3π/9*cos4π/9
=(cosπ/9*cos2π/9*cos3π/9*cos4π/9)*8(sinπ/9)/8（sinπ/9）
∵2cosπ/9*sinπ/9=sin2π/9
∴原式=(4sin2π/9*cos2π/9*cos3π/9*cos4π/9)/8（sinπ/9）
∵2sin2π/9*cos2π/9=sin4π/9
∴原式=(2sin4π/9*cos3π/9*cos4π/9)/8（sinπ/9）
=(cos3π/9*sin8π/9)/8（sinπ/9）
∵ sinπ/9=sin8π/9……诱导公式
∴原式=(cosπ/3*sin8π/9)/8（sinπ/9）
=(cosπ/3)/8
=1/16

x=cosπ/9*cos2π/9*cos3π/9*cos4π/9
y=sinπ/9*sin2π/9*sin3π/9*sin4π/9
则xy=[1/(2^4)]sin2π/9*sin4π/9*sin6π/9*sin8π/9
设z=sin2π/9*sin4π/9*sin6π/9*sin8π/9
因为sinπ/9*sin3π/9*sin5π/9*sin7π/9
=...

全部展开

x=cosπ/9*cos2π/9*cos3π/9*cos4π/9
y=sinπ/9*sin2π/9*sin3π/9*sin4π/9
则xy=[1/(2^4)]sin2π/9*sin4π/9*sin6π/9*sin8π/9
设z=sin2π/9*sin4π/9*sin6π/9*sin8π/9
因为sinπ/9*sin3π/9*sin5π/9*sin7π/9
=sin8π/9*sin10π/9*sin12π/9*sin14π/9
所以y=z
所以x=1/(2^4)
~~~通常~~~cosπ/(2n+1)*cos2π/(2n+1)*cos3π/(2n+1)*...*cosnπ/(2n+1)=1/(2^n),n是正整数
所以答案= 1/(2^4) = 1/16

收起

cosπ/9*cos2π/9*cos3π/9*cos4π/9
=(sinπ/9cosπ/9*cos2π/9*cos3π/9*cos4π/9) /(sinπ/9)
=1/2(sin2π/9*cos2π/9*cos3π/9*cos4π/9) /(sinπ/9)
=1/4(sin4π/9*cos3π/9*cos4π/9) /(sinπ/9)
=1/8(sin8π/9*cos3π/9) /(sinπ/9)
=1/8(sinπ/9*cos3π/9) /(sinπ/9)
=1/8*cos3π/9
=1/8*1/2
=1/16

cosπ/9*cos2π/9*cos3π/9*cos4π/9
=sinπ/9*cosπ/9*cos2π/9*cos3π/9*cos4π/9/sinπ/9
=1/2sin2π/9cos2π/9*1/2*cos4π/9/sinπ/9
=1/2*1/2*1/2*sin4π/9*cos4π/9/sinπ/9
=1/2*1/2*1/2*1/2*sin8π/9/sinπ/9
又sin8π/9=sinπ/9
故原式=1/16