Question

Express `cosx-cos5x+cos9x-cos13x` as a product of trigonometric ratios?

Answer 1

`cosx-cos5x+cos9x-cos13x=4sin2xsin7xcos4x`

Explanation:

As `cosB-cosA=2sin((A+B)/2)sin((A-B)/2)`

and `sinA+sinB=2sin((A+B)/2)cos((A-B)/2)`

`cosx-cos5x+cos9x-cos13x`

= `2sin((x+5x)/2)sin((5x-x)/2)+2sin((13x+9x)/2)sin((13x-9x)/2)`

= `2sin3xsin2x+2sin11xsin2x`

= `2sin2x(sin11x+sin3x)`

= `2sin2x(2sin((11x+3x)/2)cos((11x-3x)/2))`

= `2sin2x(2sin7xcos4x)`

= `4sin2xsin7xcos4x`