SOLUTION: solve on the interval (0, 2pi) cos(4X) - cos(6X) = 0

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Question 1006453: solve on the interval (0, 2pi)
cos(4X) - cos(6X) = 0

Found 2 solutions by harrysu321@gmail.com, ikleyn:
Answer by harrysu321@gmail.com(1) About Me  (Show Source):
Answer by ikleyn(52876) About Me  (Show Source):
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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

cos%28alpha%29 - cos%28beta%29 = -2%2Asin%28%28alpha-beta%29%2F2%29%2Asin%28%28alpha%2Bbeta%29%2F2%29

(see the lesson Addition and subtraction of trigonometric functions in this site).

In your case, it gives

cos%284X%29 - cos%286X%29 = 2%2Asin%28X%29%2Asin%285X%29,

and your equation takes the form

2%2Asin%28X%29%2Asin%285X%29 = 0.

It comes apart in two equations. First one is 

sin(X) = 0, and it produces the solutions X = +/- k%2Api, k = 0, 1, 2, . . . 

The second equations is sin(5x) = 0, and it produces the solutions X = +/- %28k%2Api%29%2F5, k = 0, 1, 2, . . . 

Of these two sequences, the second one overlays the first.

Taking into account the assigned interval, the solutions are %28k%2Api%29%2F5, k = 0, 1, 2, 3, . . . , 9.