I'm having trouble with a math problem involving identities. Any ideas?

[ShaunSown]

[FONT=&quot]So basically I have this math problem that tells me to use the identity, [/FONT]
[FONT=&quot]A^3 + B^3 = (A + B)*(A^2 - AB + B^2) [/FONT]
[FONT=&quot]to prove the following, [/FONT]
[FONT=&quot]1-((sin^2(x)*tan(x))/(tan(x) + 1)) - ((cos^2(x))/(tan(x)+1)) = sin(x)*cos(x) [/FONT]

[FONT=&quot]*Note: I just used 'x' instead of the theta symbol [/FONT]

[FONT=&quot]I kinda get how to prove the given identity, but I have no idea how I am supposed to use the given identity to prove the other one. Any advice or help would be greatly appreciated.[/FONT]

 

[Dr.Peterson]

So basically I have this math problem that tells me to use the identity,
A^3 + B^3 = (A + B)*(A^2 - AB + B^2)
to prove the following,
1-((sin^2(x)*tan(x))/(tan(x) + 1)) - ((cos^2(x))/(tan(x)+1)) = sin(x)*cos(x)

*Note: I just used 'x' instead of the theta symbol

I kinda get how to prove the given identity, but I have no idea how I am supposed to use the given identity to prove the other one. Any advice or help would be greatly appreciated.

It's generally best not to focus on trying to use a particular fact to prove something, but just to proceed as if you hadn't been given a hint until you see how it can be used! So I would just do whatever is reasonable on the left side, and see what happens.

In particular, I would combine fractions using a common denominator, and express everything in terms of sine and cosine. I think you will run into something like "sin^3(x) + cos^3(x)" sooner or later, and find that you can use the fact they reminded you of.

Try it, and show us what you did if you need more help.

 

[stapel]

So basically I have this math problem that tells me to use the identity,

A^3 + B^3 = (A + B)*(A^2 - AB + B^2)

to prove the following,

1-((sin^2(x)*tan(x))/(tan(x) + 1)) - ((cos^2(x))/(tan(x)+1)) = sin(x)*cos(x)

*Note: I just used 'x' instead of the theta symbol

I kinda get how to prove the given identity, but I have no idea how I am supposed to use the given identity to prove the other one. Any advice or help would be greatly appreciated.

Convert everything on the left-hand side to sines and cosines. Simplify each of the second and third expressions. You should end up with two fractions, having a common denominator, and having cubes in their numerators.

Note that each of these terms is subtracts from the initial 1, so you can convert the expression to something along the lines of:

. . . . .\(\displaystyle 1\, -\, \left(\dfrac{\mbox{something cubed}}{\mbox{common denominator}}\, +\, \dfrac{\mbox{something else cubed}}{\mbox{common denominator}}\right)\)

Combine the two fractions with the common denominators, and then apply the given identity to the numerator, which then will be a sum of cubes. Convert the 1 to a sum of squares, using the Pythagorean Identity. Most of the terms will cancel out, leaving you with what they've given you on the right-hand side.

If you get stuck, please reply showing all of your work, starting with the conversion of everything to sines and cosines. Thank you!