设f(x)=(sin2x+cos2x)/tanx+cosx,求最小正周期和值域,

问题描述:

设f(x)=(sin2x+cos2x)/tanx+cosx,求最小正周期和值域,

f(x)=(sin2x+cos2x)/(tanx+cotx)
=(sin2x+cos2x)/(sinx/cosx+cosx/sinx)
将右边代数式通分整理得到
f(x)=(sin2x+cos2x)*sinxcosx
=1/2sin²(2x)+1/4sin4x
=1/2(1-cos²2x)+1/4sin4x
=1/4-1/4cos4x+1/4sin4x
=1/4+1/4根号2sin(4x+π/4)
最小正周期为2π/4=π/2
显然值域为【1/4-1/4根号2,1/4+1/4根号2s】.
打字不易,额,你好,题目里没有cotx,是cosx哦,不好意思,我看错题目了,抱歉!没事实在抱歉!没事,解决了