题目

求值：(cosπ/8）^4+（cos3π/8）^4+(cos5π/8)^4+（cos7π/8）^4=
正确答案是3/2,我想知道解题过程,自己解了很多遍都不对!
谢谢!

提问时间：2020-12-30

答案

cos7π/8=-cosπ/8
cos5π/8=-cos3π/8=-sinπ/8
所以原式=(cosπ/8)^4+(sinπ/8)^4+(-sinπ/8)^4+(-cosπ/8)^4
=2[(cosπ/8)^4+(sinπ/8)^4]
=2{[(cosπ/8)^2+(sinπ/8)^2]^2-2(cosπ/8)^2(sinπ/8)^2}
=2{1^2-1/2*[2(cosπ/8)(sinπ/8)]^2}
=2{1-1/2*(sinπ/4)]^2]
=2*(1-1/2*1/2)
=3/2