Question 1006453
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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(alpha)}}} - {{{cos(beta)}}} = {{{-2*sin((alpha-beta)/2)*sin((alpha+beta)/2)}}}

(see the lesson <A HREF=http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-of-trigonometric-functions.lesson>Addition and subtraction of trigonometric functions</A> in this site).

In your case, it gives

{{{cos(4X)}}} - {{{cos(6X)}}} = {{{2*sin(X)*sin(5X)}}},

and your equation takes the form

{{{2*sin(X)*sin(5X)}}} = {{{0}}}.

It comes apart in two equations. First one is 

sin(X) = 0, and it produces the solutions X = +/- {{{k*pi}}}, k = 0, 1, 2, . . . 

The second equations is sin(5x) = 0, and it produces the solutions X = +/- {{{(k*pi)/5}}}, k = 0, 1, 2, . . . 

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

Taking into account the assigned interval, the solutions are {{{(k*pi)/5}}}, k = 0, 1, 2, 3, . . . , 9.  
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