Question

How do you find all solutions of the equation `cos(x+(3pi)/4)-cos(x-(3pi)/4)=0` in the interval `[0,2pi)`?

Answer 1

`0, pi, 2pi`

Explanation:

Use trig identity:
`cos a - cos b = - 2sin ((a + b)/2)sin ((a - b)/2)`
In this case:
(a + b) = 2x --> `sin ((a + b)/2) = sin x`
`(a - b) = (6pi)/4` --> `sin ((a - b)/2) = sin ((3pi)/4)`
The equation f(x) becomes:
`f(x) = cos (x + (3pi)/4) - cos (x - (3pi)/4) =`
`= - 2sin x.sin ((3pi)/4) = 0`
Since `sin ((3pi)/4) = sqrt2/2`, then:
`f(x) = - sqrt2.sin x = 0`
Unit circle -->
sin x = 0 --> `x = 0, x= pi, x = 2pi`
Answers for `(0, 2pi)`
`0, pi, 2pi`