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

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Answer 1 · 2018-03-01T23:53:52

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

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]}

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