Question

已知x∈(0,π/2),证明不存在m∈(0,1),能使等式cosx+msinx=m成立

Answer 1

m=cosx/(1-sinx)

=sin(x+pi/2)/[1+cos(x+pi/2)]

=2sin(x/2+pi/4)cos(x/2+pi/4)/[2cos(x/2+pi/2)]^2

=tan(x/2+pi/4)

x∈(0,π/2),x/2+pi/4∈(pi/4,pi/2),tan(x/2+pi/4)>1,与m>1矛盾,命题得证