`cos^3 (x) = cos(x)` Find all the solutions of the equation in the interval `0,2pi)`.

Textbook Question

Chapter 5, 5.3 - Problem 25 - Precalculus (3rd Edition, Ron Larson).
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gsarora17 | (Level 2) Associate Educator

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`cos^3(x)=cos(x)` 

Let `cos(x)=y`

`y^3=y`

`y^3-y=0`

`y(y+1)(y-1)=0`

solve for y,

y=0 , -1 , 1

Therefore cos(x)=0 , cos(x)=-1 and cos(x)=1

General solutions for cos(x)=0 are,

`x=pi/2+2pin , x=3pi/2+2pin`

Solutions for the range are ` 0<=x<=2pi `

`x=pi/2 , 3pi/2`

General solutions for cos(x)=-1 are,

`x=pi+2pin`

Solutions for the range `0<=x<=2pi` are

`x=pi` General solutions for cos(x)=1 are,

`x=0+2pin`

`x=0.2pi`

Solutions for x are,

`x=0,pi/2,pi,(3pi)/2,2pi`

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