Question

为什么sin（x+π/4）的平方等于cos（x-π/4）的平方？

Answer 1

法一：
sin（x+π/4）=sinxcosπ/4+cosxsinπ/4=√2/2(sinx+cosx)
cos(x-π/4)=cosxcosπ/4+sinxsinπ/4=√2/2(sinx+cosx)
所以相等。
法二：
sin（x+π/4）=sin(x+π/2-π/4）
=sin[π/2-(π/4-x)]
=cos(π/4-x)
=cos(x-π/4)
成立。

Answer 2

sin（x+π/4）=sinxcosπ/4+cosxsinπ/4=√2/2(sinx+cosx)
cos(x-π/4)=cosxcosπ/4+sinxsinπ/4=√2/2(sinx+cosx)
这种根据正余弦公式比较容易理解

Answer 3

{sin(x+π/4)}^2
={sin[(x-π/4)+π/2]}^2
={sin(x-π/4)cos(π/2)+cos(x-π/4)sin(π/2)}^2
={sin(x-π/4)·0+cos(x-π/4)·1}^2
={cos(x-π/4}^2