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

Answer 1

use th sum and diff formulas

cos(a+b)=cosacosb-sinasinb
cos(a-b)=cosacosb+sinasinb

cos(pi/3+x) - cos(pi/3-x) =
cos(pi/3)cosx - sin(pi/3)sinx - (cos(pi/3cosx+sin(pi/3)sinx)
=2sin(pi/3)sinx
=sqrt(3) sinx

Answer 2

=cos pi/3*cosx-sinpi/3 sinx -cospi/3 cos x-sin pi/3 sinx=
-sqrt(3) *sin (x)

Answer 3

cos (pi/3 + x) - cos (pi/3 - x)?

use:
cos(a +b) = cos(a) cos(b) -sin(a)sin(b)

cos(pi/3) =1/2
sin(pi/3) = sqrt(3) /2

cos(pi/3 +x) = (1/2) cos(x) - (sqrt(3) /2) sin(x) =
sin(x) ( 1/2 - sqrt(3) /2)


use:
cos(a -b) = cos(a) cos(b) + sin(a)sin(b)

cos(pi/3 -x) = (1/2) cos(x) + sqrt(3) /2) sin(x) =
sin(x) ( 1/2 + sqrt(3) /2)

cos (pi/3 + x) - cos (pi/3 - x) = sin(x) ( 1/2 - sqrt(3) /2) - sin(x) ( 1/2 + sqrt(3) /2) =

sin(x) { 1/2 - sqrt(3) /2 -1/2 - sqrt(3) /2} =
sin(x) {-2 sqrt(3)/2} = - sin(x) (sqrt(3) )