Question

Solve cos^2 x +cos^2 2x +cos^2 3x=1?

Answer 1

Please see below.

Explanation:

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`cos^2x+cos^2(2x)+cos^2(3x)=1`

`(1+cos2x)/2+(1+cos4x)/2+cos^2(3x)=1`

`1+cos2x+1+cos4x+2cos^2(3x)=2`

`2+cos2x+cos4x+2cos^2(3x)=2`

`cos2x+cos4x+2cos^2(3x)=0`

`2cos((4x+2x)/2)cos((4x-2x)/2)+2cos^2(3x)=0`

`cos3xcosx+cos^2(3x)=0`

`cos3x(cos3x+cosx)=0`

`cos3x=0, :. 3x=pi/2+-2kpi,(3pi)/2+-2kpi, :. x=pi/6+-2/3kpi,pi/2+-2/3kpi`

`cos3x+cosx=0`

`cos3x=-cosx`

`3x=pi+x+-2kpi, :. 2x=(1+-2k)pi, :. x=pi/2(1+-2k)`

`3x=pi-x+-2kpi, :. 4x=(1+-2k)pi, :. x=pi/4(1+-2k)`