Question

sin4次方θ+cos4次方θ=1,sinθ-cosθ=

Answer 1

Sin4次方θ+cos4次方θ=1
Sin4次方θ+cos4次方θ+2Sin2次方θcos2次方θ-2Sin2次方θcos2次方θ=1
(Sin2次方θ+cos2次方θ)2次方-2Sin2次方θcos2次方θ=1
1-2Sin2次方θcos2次方θ=1
2Sin2次方θcos2次方θ=1
4Sin2次方θcos2次方θ=1
(2Sinθcosθ)2次方=1
2Sinθcosθ=1
(sinθ-cosθ)2次方=1-2Sinθcosθ=1-1=0
sinθ-cosθ=0

追问

1-2Sin2次方θcos2次方θ=1
2Sin2次方θcos2次方θ=1 （应该=0吧？）

追答

对的！1-2Sin2次方θcos2次方θ=1
2Sin2次方θcos2次方θ=0
4Sin2次方θcos2次方θ=0
(2Sinθcosθ)2次方=0
2Sinθcosθ=0
(sinθ-cosθ)2次方=1-2Sinθcosθ=1-0=1
sinθ-cosθ=1或-1

Answer 2

sin^4θ+cos^4θ=1

sin^4θ+cos^4θ+2sin^2θcos^2θ-2sin^2θcos^2θ=1

(sin^2θ+cos^2θ)^2-2sin^2θcos^2θ=1

1-2sin^2θcos^2θ=1

2sin^2θcos^2θ=0

(sin2θ)^2=0

2θ=kπ

θ=kπ/2

sinθ-cosθ

=sin(kπ/2)-cos(kπ/2)

=sin(kπ/2)

=(-1)^(k+1)