Here,
#cos3u+cos6u=0#
#=>2cos((3u+6u)/2)cos((3u-6u)/2)=0#
#=>cos((9u)/2)cos((-3u)/2)=0#
#=>cos((9u)/2)=0 or cos((3u)/2)=0#
We have two options;
#(I)cos((9u)/2)=0=>(9u)/2=pi/2,(3pi)/2,(5pi)/2,(7pi)/2,(9pi)/2...#
#=>9u=pi,3pi,5pi,7pi,9pi,...#
#=>u=pi/9,(3pi)/9,(5pi)/9,(7pi)/9,(9pi)/9,...#
But, #uin [0,(2pi)/3)#i.e. #uin[0^circ,120^circ)#
#:.u=pi/9,(3pi)/9,(5pi)/9, are # possible.
#:.u=pi/9,pi/3,(5pi)/9.#
#(II)cos((3u)/2)=0=>(3u)/2=pi/2,(3pi)/2,(5pi)/2,(7pi)/2,(9pi)/2...#
#=>3u=pi,3pi,5pi,7pi,9pi,...#
#=>u=pi/3,(3pi)/3,(5pi)/3,...#
But #uin [0,(2pi)/3)=>u=pi/3 #, is possible.
From #(I) and(II)# we have #u=pi/9,pi/3,(5pi)/9.#