同角三角函数的基本关系.

题目

同角三角函数的基本关系.
若sinx+cosx=根号下2,那么(sin^4)x+(cos^4)x的值为( )

答案

方法1: 因为((sin^2)x+(cos^2)x)^2=(sin^4)x+(cos^4)x+2*(sin^2)x (cos^2)x
由 (sinx+cosx)^2=(sin^2)x+(cos^2)x+2*sinx cosx,即
sinx cosx = ((sinx+cosx)^2-=(sin^2)x+(cos^2)x)/2 =(2-1)/2=1/2
所以(sin^4)x+(cos^4)x = ((sin^2)x+(cos^2)x)^2- 2*(sin^2)x (cos^2)x=1-2*(1/2)^2=1/2
方法2:取特值取sinx=cosx=√2/2,即(sin^4)x+(cos^4)x=(√2/2)^4+(√2/2)^4=1/2