作业帮已知函数f(x)=(sin2x+cos2x)/(tanx+cotx),求f(x)的值域.和周期
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已知函数f(x)=(sin2x+cos2x)/(tanx+cotx),求f(x)的值域.和周期
已知函数f(x)=(sin2x+cos2x)/(tanx+cotx),求f(x)的值域.和周期
已知函数f(x)=(sin2x+cos2x)/(tanx+cotx),求f(x)的值域.和周期
f(x)=(sin2x+cos2x)/(tanx+cotx)
=(sin2x+cos2x)/(sinx/cosx+cosx/sinx)
=(sin2x+cos2x)/[(si'n^x+cos^x)/sinxcosx]
=(sin2x+cos2x)/(1/sinxcosx)
=(sin2x+cos2x)sinxcosx
=(1/2)(sin2x+cos2x)sin2x
=(1/4)[2sin^(2x)+2cos2xsin2x]
=(1/4)[1-cos4x+sin4x]
=1/4-根号2*sin(4x-∏/4]/4 ,x≠k∏/2, 4x≠2k∏
-1<=sin(4x-∏/4]<=1
根号2/4<=-根号2*sin(4x-∏/4]/4<=根号2/4,
所以(1-根号2)/4<=1/4-根号2*sin(4x-∏/4]/4<=(1+根号2)/4
所以f(x)的值域[(1-根号2)/4,(1+根号2)/4]
周期2∏/4=∏/2
tanx+cotx
=sinx/cosx+cosx/sinx
=（sin平方+cos平方）/(sinxcosx)
=1/sinxcosx
f(x)=(sin2x+cos2x)/(tanx+cotx)
=(sin2x+cos2x)/(1/sinxcosx)
=(sin2x+cos2x)*sinxcosx
=(sin2x+...
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tanx+cotx
=sinx/cosx+cosx/sinx
=（sin平方+cos平方）/(sinxcosx)
=1/sinxcosx
f(x)=(sin2x+cos2x)/(tanx+cotx)
=(sin2x+cos2x)/(1/sinxcosx)
=(sin2x+cos2x)*sinxcosx
=(sin2x+cos2x)*(1/2)*sin2x
=(1/2)*sin2x平方＋(1/2)*sin2xcos2x
=(1/4)*(2sin2x平方－1＋1)＋(1/4)*sin4x
=(1/4)*(-cos4x+1)+(1/4)*sin4x
=1/4+(1/4)*(sin4x-cos4x)
=1/4+(根号2/4)*sin(4x-派/4)
因此，f（x）的值域为[1/4-根号2/4，1/4+根号2/4]
周期为2派/4＝派/2
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