Need some math help

[spaceghost21]

We're working on Trigonometric Identities and Equations, and I pretty much have it down, but one problem has me stumped.

Directions: Prove the equation is an identity.

csc^6 x - cot^6 = 1 + 3 csc^2x * cot^2 x

At first I thought it might have something to do with x^3 - a^3 = (x - a) (x^2 + ax + a^2):

(csc^2 x)^3 - (cot^2 x)^3 = (csc^2 x - cot^2 x) (csc^4 x + csc^2 x * cot^2 + cot^4)

(csc^2 x - cot^2 x) turns into 1, but then I'm stuck with (csc^4 x + csc^2 x * cot^2 + cot^4).

Am I on the right track? Could anyone help me get started in the right direction?

 

[Nab]

csc^6 x - cot^6 = 1 + csc^2x * cot^2 x


the csc^2x * cot^2 x is supposed to be multiplied first right? i mean...just making sure you're not forgetting paranthesis

 

[spaceghost21]

Right.

I just noticed however, I forgot to add the 3 before csc^2x * cot^2 x .
My mistake, I'm getting frustrated with this. 😀

Thanks in advance for any help.

 

[Nab]

ewww i got something wrong, i'll try again later tonight....

 

[Zerhyn]

Is this what it is supposed to look like...

this

 

[Darien]

Originally posted by: spaceghost21
(csc^2 x)^3 - (cot^2 x)^3 = (csc^2 x - cot^2 x) (csc^4 x + csc^2 x * cot^2 + cot^4)

(csc^2 x - cot^2 x) turns into 1, but then I'm stuck with (csc^4 x + csc^2 x * cot^2 + cot^4).

So you have:

csc^6 - cot^6 = csc^4 + csc^2 * cot^2 + cot^4

Since you want to eventually end up with an expression containing a 1 and a 3 * csc^2 * cot^2, try adding what you need (add by 0)

csc^6 - cot^6 = csc^4 + csc^2 * cot^2 + cot^4 - 2csc^2 * cot^2 + 2csc^2 * cot^2
csc^6 - cot^6 = csc^4 + (3csc^2 * cot^2) + cot^4 - 2csc^2 * cot^2

You can then rearrange the terms on the right side:
csc^6 - cot^6 = csc^4 - 2csc^2 * cot^2 + cot^4 + (3csc^2 * cot^2)
csc^6 - cot^6 = (csc^2 - cot^2)^2 + (3csc^2 * cot^2)
csc^6 - cot^6 = 1 + 3csc^2 * cot^2

 

[spaceghost21]

Originally posted by: Darien

Originally posted by: spaceghost21
(csc^2 x)^3 - (cot^2 x)^3 = (csc^2 x - cot^2 x) (csc^4 x + csc^2 x * cot^2 + cot^4)

(csc^2 x - cot^2 x) turns into 1, but then I'm stuck with (csc^4 x + csc^2 x * cot^2 + cot^4).

So you have:

csc^6 - cot^6 = csc^4 + csc^2 * cot^2 + cot^4

Since you want to eventually end up with an expression containing a 1 and a 3 * csc^2 * cot^2, try adding what you need (add by 0)

csc^6 - cot^6 = csc^4 + csc^2 * cot^2 + cot^4 - 2csc^2 * cot^2 + 2csc^2 * cot^2
csc^6 - cot^6 = csc^4 + (3csc^2 * cot^2) + cot^4 - 2csc^2 * cot^2

You can then rearrange the terms on the right side:
csc^6 - cot^6 = csc^4 - 2csc^2 * cot^2 + cot^4 + (3csc^2 * cot^2)
csc^6 - cot^6 = (csc^2 - cot^2)^2 + (3csc^2 * cot^2)
csc^6 - cot^6 = 1 + 3csc^2 * cot^2

Thank you very much. I was about to go insane.

 

[Nab]

ah, looks like someone got it 🙂