Question

cosπ/5cos2π/5cos3π/5cos4π/5

Answer 1

解法一：
cosπ/5cos2π/5cos3π/5cos4π/5
=sinπ/5cosπ/5cos2π/5cos3π/5cos4π/5/sinπ/5
=1/2sin2π/5cos2π/5cos3π/5cos4π/5/sinπ/5
=1/4sin4π/5cos3π/5cos4π/5/sinπ/5
=1/8sin8π/5cos3π/5/sinπ/5
=1/8sin(π+3π/5)cos3π/5/sinπ/5
=-1/8sin3π/5cos3π/5/sinπ/5
=-1/16sin6π/5/sinπ/5
=-1/16sin(π+π/5)/sinπ/5
=1/16sinπ/5/sinπ/5
=1/16

解法二
cosπ/5cos2π/5cos3π/5cos4π/5
=cosπ/5cos2π/5cos(π-2π/5)cos(π-π/5)
=cosπ/5cos2π/5cos2π/5cosπ/5
=cos^2(π/5)cos^2(2π/5)
=sin^2(π/5)cos^2(π/5)cos^2(2π/5)/sin^2(π/5)
=1/4sin^2(2π/5)cos^2(2π/5)/sin^2(π/5)
=1/16sin^2(4π/5)/sin^2(π/5)
=1/16sin^2(π+π/5)/sin^2(π/5)
=1/16sin^2(π/5)/sin^2(π/5)
=1/16