x是（0,π/2）若sinxcosx=1/2,求1/(1+sinx) +1/(1+cosx)的值

题目

x是（0,π/2）若sinxcosx=1/2,求1/(1+sinx) +1/(1+cosx)的值

答案

由sin(x)cos(x)=1/2知
(sin(x)+cos(x))^2
=sin^2(x)+cos^2(x)+2sin(x)cos(x)
=1+2*1/2
=2,
所以sin(x)+cos(x)=Sqrt(2)（由x在第一象限知sin(x)>0,cos(x)>0）,
所以
1/(1+sin(x)) +1/(1+cos(x))
=[(1+sin(x))+(1+cos(x))]/[(1+sin(x)) (1+cos(x))]
=[2+sin(x)+cos(x)]/[1+sin(x)+cos(x)+six(x)cos(x)]
=(2+Sqrt(2))/(1+Sqrt(2)+1/2)
=(4+2*Sqrt(2))/(3+2*Sqrt(2))
=(4+2*Sqrt(2))(3-2*Sqrt(2))
=4-2*Sqrt(2)