Question

How do you find all solutions of the equation `cos(x+pi/4)-cos(x-pi/4)=1` in the interval `[0,2pi)`?

Answer 1

`(5pi)/4, (7pi)/4`

Explanation:

Use the trig identity:
`cos a - cos b = - 2sin ((a + b)/2).sin ((a - b)/2)`
In this case:
a = (x + pi)/4, and b = (x - pi/4)
(a + b) = 2x --> (a + b)/2 = x
(a - b) = pi/2 --> (a - b)/2 = pi/4
There for:
cos (x + pi/4) - cos (x - pi/4) = - 2sin x.sin (pi/4) = 1
Trig table gives --> `sin (pi/4) = sqrt2/2`, then -->
`- sqrt2.sin x = 1` --> `sin x = - 1/sqrt2 = - sqrt2/2`
Trig table and unit circle give 2 solutions:
`x = - pi/4` and `x = pi + pi/4 = (5pi)/4`
`x = - pi/4` is co-terminal to `x = (7pi)/4`.
Answers for `(0, 2pi)`
`(5pi)/4, (7pi)/4`