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Question

The number of distinct solutions of the equation 54cos22x+cos4x+sin4x+cos6x+sin6x=2 in the interval [0,2π] is___

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Solution

54cos22x+cos4x+sin4x+cos6x+sin6x=2
54cos22x+(sin2x+cos2x)22sin2xcos2x
+(sin2x+cos2x)33sin2xcos2x(sin2x+cos2x)=254cos2x+112sin22x+134sin22x=2
cos22x=sin22x
tan22x=1
tan2x=±1
2x=nπ±π4,nZ
x=(4n±1)π8,nz
x=π8,3π8,5π8,7π8,9π8,11π8,13π8,15π8


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