cos(5 theta)=Re[e^(5 i theta)]=Re [(e^(i theta))^5]

......=Re[ (cos(theta)+isin(theta))^5]

......=Re[c^5+5 i s c^4-10 s^2 c^3 -10 i s^3 c^2 +5 s^4 c + i s^5]

......=c^5 - 10(1-c^2)c^3 + 5(1-c^2)^2 c

......=16 c^5 -20 c^3 + 5 c

Thence the roots of:

x(16x^4-20x^2+5)=0

are the roots of cos(5 x)=0, and the given roots are distinct roots of this

equation.

RonL