Question

How do you simplify `cos (pi/3 + x) - cos (pi/3 - x)`?

Answer 1

The simplified expression is `-sqrt(3)*sinx`

Explanation:

Use the cosine sum and difference formulae:

`cos(alpha+beta)=cos(alpha)*cos(beta)-sin(alpha)*sin(beta)`

`cos(alpha-beta)=cos(alpha)*cos(beta)+sin(alpha)*sin(beta)`

I color-coded both parts of the expression so it's easier to follow:

`color(red)(cos(pi/3+x))-color(blue)(cos(pi/3-x))`

`color(red)((cos(pi/3)*cosx-sin(pi/3)*sinx))-color(blue)((cos(pi/3)*cosx+sin(pi/3)*sinx))`

`color(red)((1/2*cosx-sqrt3/2*sinx))-color(blue)((1/2*cosx+sqrt3/2*sinx))`

`color(red)((cosx/2-(sqrt3*sinx)/2))-color(blue)((cosx/2+(sqrt3*sinx)/2))`

`color(red)(cosx/2-(sqrt3*sinx)/2)color(blue)(-cosx/2-(sqrt3*sinx)/2)`

`cancel(color(red)(cosx/2)color(blue)(-cosx/2))color(red)(-(sqrt3*sinx)/2)color(blue)(-(sqrt3*sinx)/2)`

`color(red)(-(sqrt3*sinx)/2)color(blue)(-(sqrt3*sinx)/2)`

`color(black)((color(red)(-sqrt3*sinx)color(blue)(-sqrt3*sinx))/2)`

`(-color(red)cancel(color(black)2)sqrt3*sinx)/color(red)cancel(color(black)2)`

`-sqrt3*sinx`

You can verify this answer by graphing both `cos(pi/3+x)-cos(pi/3-x)` and `-sqrt3*sinx` and seeing that they are the same graph:

graph{-sqrt(3)*sinx [-10, 10, -5, 5]}