Solve for x and y. cos y=(x+x^-1)/2
We'll apply the inequality between the arithmetical mean and geometrical mean:
for a>0, b>0, then
Particularly, fora>0, then
a+(1/a)>2sqrt(a*1/a)=2, and the equality takes place if only a=1.
If a<0, then a+(1/a)<-2 and the equality takes place if only a=-1.
We know that: -2<2cos y<2
From all the above, the result is if x>0, then
x+(1/x)>2>2cos y (Note: we'll write x^-1 = 1/x)
We can have solutions if only
2 cos y=2 if x=1
y=2k*pi, where k is in Z set.
If x<0, then x+(1/x)<-2<2 cos y
We can have solutions if only x+(1/x)=-2 and 2 cos y=-2, meaning if x=-1 and y= pi + 2k*pi
The roots of the equation x+(1/x)=2 cos y are:
x=1 and y=2k*pi
x=-1 and y=pi+2k*pi, where k is integer