2010/046) If 2cos(a)=x+1/x and 2cos(b)=y+1/y.so then what is the value of [x/y+y/x]

x^2 - 2x cos a + 1 = 0

so x = (2 cos a +/- sqrt(4 cos^2 a - 4))/2 using discriminant
= cos a +/- sqrt(cos^2 a - 1)
= cos a +/- sqrt(- sin ^2a)
= cos a +/- i sin a = e^ia or e^-ia

taking e^ia

similarly y = e^ib

x/y = e^i(a-b) = cos(a-b) + i sin (a-b)

y/x = e^-i(a-b) = cos(a-b) - i sin (a-b)

adding we get x/y + y/x = 2 cos (a-b)