Question

How do you use sum to product formulas to write the sum or difference `cos(x+pi/6)-cos(x-pi/6)` as a product?

Answer 1

`-2sinx sin(pi/6)`

Explanation:

use the difference formula: `cosu - cosv = -2sin((u+v)/2) sin((u-v)/2)`

Let `u = x + pi/6` and `v = x - pi/6`

`(u+v)/2 = (x + pi/6 + x - pi/6)/2 = (2x)/2 = x`

`(u-v)/2 = (x + pi/6 - (x - pi/6))/2 = (x + pi/6 - x + pi/6)/2`

`= (2pi)/6 * 1/2 = pi/6`

So, `cos(x + pi/6) - cos (x - pi/6) = -2 sinx sin(pi/6)`