已知sinα+cosα =√2/2,求1/(sin2α)+1/(cos2α)的值
问题描述：
已知sinα+cosα =√2/2,求1/(sin2α)+1/(cos2α)的值
答
∵sinα+cosα =√2/2
且(sinα+cosα)^2 =sin^2α+cos^2α+2sinαcosα=1/2
∴sin2α=2sinαcosα=-1/2
∵sin^2α+cos^2α =1
且sinα+cosα =√2/2
∴α=-π/12+2kπ or 7π/12+2kπ
∴cos2α=√3/2 or -√3/2
∴等式=(2√3-6)/3 or (2√3+6)/3
答
sinα+cosα =√2/2,
两边平方,
1+sin2α=1/2,
sin2α=-1/2,
cos2α=±√3/2,
1/(sin2α)+1/(cos2α)=-2±2√3/3.