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solve on the interval (0, 2pi)
cos(4X) - cos(6X) = 0
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Use the general formula for subtraction of cosines
- =
(see the lesson Addition and subtraction of trigonometric functions in this site).
In your case, it gives
- = ,
and your equation takes the form
= .
It comes apart in two equations. First one is
sin(X) = 0, and it produces the solutions X = +/- , k = 0, 1, 2, . . .
The second equations is sin(5x) = 0, and it produces the solutions X = +/- , k = 0, 1, 2, . . .
Of these two sequences, the second one overlays the first.
Taking into account the assigned interval, the solutions are , k = 0, 1, 2, 3, . . . , 9.