Write cos (3x) - cos x as a product

-2sin (x) * sin (2x)

or any equivalent form, such as 4cos (x) [cos² (x) - 1]

Step-by-step explanation:

simplify cos (3x) first:

cos (3x) = cos (2x+x) = cos (2x) cos (x) - sin (2x) sin (x)

using trig identities

= [2cos² (x) - 1]cos (x) - [2sin (x) cos (x) ]sin (x)

= 2cos³ (x) - cos (x) - 2sin² (x) cos (x)

substituting using trig identity sin² (x) + cos² (x) = 1

2cos³ (x) - cos (x) - 2[1-cos² (x) ]cos (x)

2cos³ (x) - cos (x) - 2cos (x) + 2cos³ (x)

4cos³ (x) - 3cos (x)

remember this cos (3x), we still have to subtract cos (x)

4cos³ (x) - 3cos (x) - cos (x) = 4cos³ (x) - 4cos (x)

we can factor 4cos (x) to write this as a product of:

4cos (x) [cos² (x) - 1]

further simplification if you want

trig identity sin² (x) + cos² (x) = 1

simplifying: sin² (x) = 1-cos² (x)

simplifying: - sin² (x) = cos² (x) - 1

4cos (x) [cos² (x) - 1]

4cos (x) [-sin² (x) ]

-4cos (x) sin² (x)

trig identity: sin (2a) = 2cos (a) sin (a)

-2sin (x) * 2cos (x) sin (x)

-2sin (x) * sin (2x)