Question

已知sinx+cosx=根号2/2,求sinx的四次方+cosx的四次方

Answer 1

设a=sinx,b=cosx,题中的隐含条件是a^2+b^2=1

由a+b=√2/2得(a+b)^2=1/2,a^2+2ab+b^2=1/2,2ab=-1/2,ab=-1/4

(a^2+b^2)^2=1,又

a^4+2a^2b^2+b^4=a^4+b^4+2(ab)^2=a^4+b^4+1/8

∴a^4+b^4=1-1/8=7/8,即sin^4 x+cos^4 x=7/8