Question

已知α为锐角,且sinαcosα=1/2,则1/(1+sinα)+1/(1+cosα)=____

写下详细过程 谢谢

Answer 1

∵α为锐角
∴sinα＞0，cosα＞0
∴sinα+cosα＞0
∴sinα+cosα = √(sinα+cosα)^2 = √(sin^2α+cos^2α+2sinαcosα) = √(1+2sinαcosα) = √(1+2*1/2) = √2

∴1/(1+sinα)+1/(1+cosα)
= (1+cosα+1+sinα)/{(1+sinα)(1+cosα) }
= (2+sinα+cosα)/(1+sinα+cos+sinαcosα)
=(2+√2)/(1+√2+1/2)
=2(2+√2)/(3+2√2)
=2(2+√2)(3-2√2)/{(3+2√2)(3-2√2)}
=4-2√2