Question

若sinθ+cosθ=根号2 求log1/2(sinθ)·log1/2(cosθ)

Answer 1

sinθ+cosθ=根号2

两边平方,1+2sinθcosθ=2

2sinθcosθ=1

2sinθ(根号2－sinθ)=1

2sinθ^2-2根号2sinθ+1=0

sinθ＝根号2/2

cosθ=根号（1-sinθ^2）=根号2/2

log1/2(sinθ)·log1/2(cosθ)

＝(log1/2(根号2/2))^2

＝(log1/2 (1/2)^(1/2))^2

=1/4