cos5x - cos7x = 0

Question

cos5x - cos7x = 0

tkhunny · Answer

You MUST write clearly and unambiguously.

$\cos^{5}(x) - \cos^{7}(x)\;or\;\cos(5x) - \cos(7x)$

Parentheses are amazign things!

anonymous · Answer

but i thought cos(5x) is cos of 5 times x, not the whole cosx to the power of 5

tkhunny · Answer

WITH the parentheses that is clear.  You did not supply the parentheses initially.

So then, we have $\cos(5x) - \cos(7x)$?

anonymous · Answer

yeah  =0

tkhunny · Answer

Can you see at least one obvious solution?

anonymous · Answer

are cos(5x) and cos(7x) equal $$\cos (5x) = \cos(7x)$$

tkhunny · Answer

Expand $\cos(7x) = \cos(5x+2x) = \cos(5x)\cos(2x) - \sin(5x)\sin(2x)$
Factor $\cos(5x)[1 - cos(2x)]$
Substitute $\cos(2x) = 1-2\sin^{2}(x)$
Substitute $\sin(2x) = 2\sin(x)\cos(x)$

Factor out $2\sin(x)$ and observe solutions $\sin(x) = 0$

Contract $\cos(5x)\sin(x)+\sin(5x)\cos(x) = \sin(6x)$

And the remainder of the solutions are relatively simple after all that!

tkhunny · Answer

Right x = 0 is a quick and obvious solution.  Now, follow those steps carefully.  Use your best algebra.  Go get the rest of them.

anonymous · Answer

Thanks this was really helpful

tkhunny · Answer

It's pretty complicated.   If you did not sleep through your algebra, you should be able to wade through it.