Saturday, 8 April 2017

trigonometry - How prove this equation has only one solution cos(2x)+cosxcdotcos(sqrt(pi3x)(pi+x))=0


Let x(0,π3].
Show that this equation
cos(2x)+cosxcos((π3x)(π+x))=0
has a unique solution x=π3




I try to the constructor f(x)=cos(2x)+cosxcos((π3x)(π+x))=0,f(π3)=0but I use found this function is not a monotonic function

see wolframpha



Now the key How to prove this function f(x) in (0,π3) has no solution, since
f(π6)=12+13cos(1273π)=0.138<0
in other words, how to prove that
f(x)<0,x(0,π3).

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