Question

Simplify `cos x - cos 2x + sin 2x - sin 3x =`?

Answer 1

Use these trig identities:
`cos a - cos b= - 2sin ((a + b)/2)sin ((a - b)/2)`
`sin a - sin b = 2cos ((a + b)/2)sin ((a - b)/2)`
In this case:
`cos x - cos 2x = 2sin ((3x)/2)sin (x/2)` (1)
`sin 2x - sin 3x = -2cos ((5x)/2)sin (x/2)` (2)
Add (1) and (2).
Put `sin (x/2)` into common factor, we get:
`cos x - cos 2x + sin 2x - sin 3x =`
`= 2sin (x/2)[sin ((3x)/2) - cos ((5x)/2)]`