Simplifying ANOTHER very anoying equation

[Titandwedebil]

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".

 

[dextercioby]

Think of (a+b)^2 expansion...

 

[Titandwedebil]

What's that?

 

[dextercioby]

Well, (a+b)^2 = a^2 +b^2 + 2ab. I think this algebraic identity could be useful...Oh, and 1^2 =1.

 

[eumyang]

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
$$\cos^4 \theta + \sin^4 \theta = 1$$
.

Think of (a+b)^2 expansion...

What's that?

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

 

[Ray Vickson]

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

 

[HallsofIvy]

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.