已知锐角α满足sin2α=1/4,则1/(1+sinα)+1/(1+cosα)=?
问题描述：
已知锐角α满足sin2α=1/4,则1/(1+sinα)+1/(1+cosα)=?
答
sin2α=1/4
1+2sin2α=1+1/4
sin^2α+cos^2α+2sinαcosα =5/4
(sinα+cosα)^2=5/4
α为锐角
sinα+cosα=根号5/2.(1)
sin2α=1/4
2sinαcosα=1/4
sinαcosα=1/8.(2)
1/(1+sinα)+1/(1+cosα)
= (1+cosα+1+sinα) / {(1+sinα)(1+cosα)}
= (2+sinα+cosα) / (1+sinα+cosα+sinαcosα)
= (2+根号5/2) / (1+根号5/2+1/8)
= 4(4+根号5) / （9+4根号5)
= 4(4+根号5)(9-4根号5) / {9^2-(4根号5)^2}
= 4(16-7根号5）