`cos(x + pi/4) - cos(x - pi/4) = 1` Find all solutions of the equation in the interval [0, 2pi).

Monday, October 8, 2012

`cos(x+pi/4)-cos(x-pi/4)=1 , 0<=x<=2pi`

We will use the following identity,

`cos(A+B)=cosAcosB-sinAsinB`

`cos(x+pi/4)-cos(x-pi/4)=1`

`rArr (cos(pi/4)cos(x)-sin(pi/4)sin(x))-(cos(pi/4)cos(x)+sin(pi/4)sin(x))=1`

`rArr(cos(x)-sin(x))/sqrt(2)-(cos(x)+sin(x))/sqrt(2)=1`

`rArr(cos(x)-sin(x)-cos(x)-sin(x))/sqrt(2)=1`

`rArr(-2sin(x))/sqrt(2)=1`

`rArrsin(x)=-1/sqrt(2)`

General solutions are ,

`x=(5pi)/4+2pin , x=(7pi)/4+2pin`

solutions for the range `0<=x<=2pi` are,

`x=(5pi)/4 , x=(7pi)/4`

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