求证cos4°*cos8°*cos12°* …*cos88°==1/2^22

证明 根据己知恒等式:
cosa*cos(60°-a)*cos(60°+a)=(cos3a)/4.
cos4°*cos8°*cos12°* …*cos88°=[cos4°*cos56°*cos64°]*[cos8°*cos52°*cos68°]*[cos12°*cos48°*cos72°]*[cos16°*cos44°*cos76°]*[cos20°*cos40°*cos80°]*[cos24°*cos36°*cos84°]*[cos28°*cos32°*cos88°]*cos60°
=(1/2)^15*[cos12°*cos24°*cos36°*cos48°*cos60°*cos72°*cos84°]
=(1/2)16*[cos12°*cos48°*cos72°]*[cos24°*cos36°*cos84°]
=(1/2)^20*[cos36°*cos72°]=(1/2)^20*[cos36°*sin18°]
=(1/2)22*[4*sin18°*cos18°*cos36°/cos18°]
=(1/2)^22*[sin72°/cos18°]=(1/2)^22. 证毕。

cos4°*cos8°*cos12°* …*cos88°
=sin4cos4°*cos8°*cos12°* …*cos88°/sin4
=1/2^22sin176/sin4
又sin176＝sin（180－176）＝sin4
所以cos4°*cos8°*cos12°* …*cos88°==1/2^22