Simplifying ANOTHER very anoying equation

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Homework Help Overview

The discussion revolves around simplifying the expression (cos4x) + 2(cos2x)(sin2x) + (sin4x), which involves trigonometric identities and algebraic manipulation.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the use of Pythagorean identities and the expansion of binomials. There is a suggestion to rewrite the expression in terms of sine and cosine, and some participants question the validity of applying the Pythagorean identity directly to the fourth powers of sine and cosine.

Discussion Status

The discussion is active, with participants sharing different approaches and questioning assumptions. Some guidance has been offered regarding algebraic identities, but there is no clear consensus on the best method to simplify the expression.

Contextual Notes

Participants are working under the constraints of homework rules, which may limit the types of solutions or methods they can use. There is also a mention of a specific expected answer, which adds pressure to find a valid simplification.

Titandwedebil
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Homework Statement


(cos4x) + 2(cos2x)(sin2x) + (sin4x)


Homework Equations


All Pythagorean, reciprocal, and quotient identities.


The Attempt at a Solution


Okay, so I thought that maybe (cos4x) + (sin4x) might be the same as its Pythagorean identity (which would make that mess just "1"; giving me...

2(cos2x)(sin2x) + 1

From there I haven't the slightest idea on what to do. The answer to this one is supposed to be just "1".
 
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What's that?
 
Well, (a+b)^2 = a^2 +b^2 + 2ab. I think this algebraic identity could be useful...Oh, and 1^2 =1.
 
Titandwedebil said:
Okay, so I thought that maybe (cos4x) + (sin4x) might be the same as its Pythagorean identity (which would make that mess just "1"

No, that's not right. Just because
\cos^2 \theta + \sin^2 \theta = 1
doesn't necessarily mean that
\cos^4 \theta + \sin^4 \theta = 1
.

dextercioby said:
Think of (a+b)^2 expansion...
Titandwedebil said:
What's that?

We call this the "Square of a Binomial Pattern," typically learned in high school algebra.
 
Titandwedebil said:

Homework Statement


(cos4x) + 2(cos2x)(sin2x) + (sin4x)


Homework Equations


All Pythagorean, reciprocal, and quotient identities.


The Attempt at a Solution


Okay, so I thought that maybe (cos4x) + (sin4x) might be the same as its Pythagorean identity (which would make that mess just "1"; giving me...

2(cos2x)(sin2x) + 1

From there I haven't the slightest idea on what to do. The answer to this one is supposed to be just "1".

Write s = sin(x) and note that cos^2(x) = 1-s^2, so you have (1-s^2)^2 + 2*(1-s^2)*s^2 + s^4. Expand it out.

RGV
 
Ray Vickson said:
Write s = sin(x) and note that cos^2(x) = 1-s^2, so you have (1-s^2)^2 + 2*(1-s^2)*s^2 + s^4. Expand it out.

RGV
That looks like the hard way to do it! DexterCioby's idea is best.
 

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