Let x∈(0,π3].
Show that this equation
cos(2x)+cosx⋅cos(√(π−3x)(π+x))=0
has a unique solution x=π3
I try to the constructor f(x)=cos(2x)+cosx⋅cos(√(π−3x)(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+1√3cos(12√73π)=−0.138⋯<0
in other words, how to prove that
f(x)<0,∀x∈(0,π3).
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