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Topic review (newest first)

harleyd2900
2005-12-15 14:28:45

How would you work it from here?

cos²5x - cos²x = (-sin4x)(sin6x)
(cos5x - cosx)(cos5x + cosx)

?

harleyd2900
2005-12-15 13:52:01

Are there any "more basic" ways you whould attempt doing it.  Or without using that specific formula. IDK what I am asking. Maybe he didn't teach us that because its only a Pre-Cal class.

Ricky
2005-12-15 13:36:04

It's a standard trig identity yes.

harleyd2900
2005-12-15 13:22:21

Its funny he has never shown us that formula sinAsinB = 1/2(cos(A-B) - cos(A+B)). Is it a given?

Ricky
2005-12-15 13:06:49

Not a problem.  Mostly all you have to do is work backwards:

cos²(5x) - cos²(x) = 1/2(2cos²(5x)-1+1 - (2cos²(x) - 1 + 1) )
                           = 1/2(cos²(10x) - cos²(2x) )

Since 10x = 6x - (-4x) and 2x = 6x + (-4x):
                 
                           = sin(6x)sin(-4x)
                           = -sin(4x)sin(6x)

I normally do a lot more steps than needed, just so others can easily follow along.

harleyd2900
2005-12-15 12:46:42

darn that is great. However he said it could be proven in 4-5 steps by only working the left side.... Not to be picky, but you wouldn't happen to know how do do it like that?

PS: (Sorry I am being picky, I have a test on this type of problem tomarow.)

Thanks so much for your help

Ricky
2005-12-15 12:24:22

http://www.pen.k12.va.us/Div/Winchester/jhhs/math/lessons/trig/ident2.html

sinAsinB = 1/2(cos(A-B) - cos(A+B))
-sinA = sin(-A)
cos(2A) = cos²A - sin²A

-sin(4x) = sin(-4x)

sin(-4x) * sin(6x) = 1/2(cos(-4x - 6x) - cos(-4x + 6x)) = 1/2(cos(-10x) - cos(2x))

1/2(cos(2 * -5x) - cos(2 * x)) = 1/2(cos²(-5x) - sin²(-5x) - ( cos²(x) - sin²(x) ) )

Since sin²(-5x) = 1 - cos²(-5x):

1/2(cos²(-5x) - (1 - cos²(-5x)) - (cos²(x) - (1 - cos²(x)) ) )

1/2( 2cos²(-5x) - 1 -2cos²(x) + 1) = 1/2(2cos²(-5x) - 2cos²(x)) = cos²(-5x) - cos²(x)

And finally, since cos(-A) = cos(A):

cos²(5x) - cos²(x)

harleyd2900
2005-12-15 11:47:07

You can use the identities. The words he really used were varifying Trigonometric Idenities. "Proving" that one side equals the other side, or the working of one side to equal the other.

Hope that helped.

Ricky
2005-12-15 07:21:48

The title is "Proving an Idenity" so does this mean you are not allowed to use trig identities?  As that would be begging the question.

Hard Prove
2005-12-15 06:14:57

My Pre-Cal teacher gave us this problem today. I have worked on it for a very long time and have goten no where. I was wondering if anyone had any ideas.

cos²5x - cos²x = -sin4x  x  sin6x


If that is not clear


cos sqared of five x minus cos squared of x =  negative sin of 4x times sin of 6x




Thanks a lot for your help,

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