Solving Simple Problem Check: Sin^4X - Cos^4X

[I'm]

Homework Statement

Sin ^{4}X - cos^{4}X

Homework Equations

The Attempt at a Solution

I'm just checking if I can switch the sin and cos at the beginning to make:

-Cos^{4}X + Sin^{4}X

Then multiply by -1 to basically switch the the Sin and Cos around?

I'm supposed to simplify, and if all that works I get Cos2x. Is this right?

 

[bucher]

You've almost got it. I would multiply by 1 (-1/-1). Use one of the -1s to do your simplification. The other -1 would just make the answer -cos(2x). I'm pretty sure this will work but I'd double check this.

 

[I'm]

I understand from the point of a variation of the double angle formulas, but can you expliain your reasoning a bit to me?

I get two different answers from two different methods. It worries me hah.

 

[qntty]

you can't multiply an expression by something to simplify it unless you are multiplying by 1. Try factoring and substituting in identities just like you tryed the first time but without multiplying by -1.

 

[I'm]

Thanks, I guess I just have to memorize that.

Thank you !

 

[Дьявол]

sin^4x-cos^4x=(sin^2x)^2-(cos^2x)^2=(sin^2x-cos^2x)(sin^2x+cos^2x)

Is it clear now?

 

[Mark44]

I'm just checking if I can switch the sin and cos at the beginning to make:

-Cos^{4}X + Sin^{4}X

Then multiply by -1 to basically switch the the Sin and Cos around?

I'm supposed to simplify, and if all that works I get Cos2x. Is this right?

Because a + b = b + a for any real number a and b, you can rewrite -cos⁴(x) + sin⁴(x) as sin⁴(x) + (-cos⁴(x)). The latter expression is also equal to sin⁴(x) - cos⁴(x).

 

[I'm]

sin^4x-cos^4x=(sin^2x)^2-(cos^2x)^2=(sin^2x-cos^2x)(sin^2x+cos^2x)

Is it clear now?

Yes, I did that and sin ^2x + Cos ^2X = 1

so that leaves me with sin^2x - cos^2x, which I have to simplify.

I turned that into (- cos^2x + Sin^2x) for simplification purposes.

Then I multiplied that whole thing by -1/-1

Which gave me Cos ^2x - Sin ^2x, which simplifies to Cos 2x. Divided by -1, is -cos2x, which is my answer.

Correct?

 

[Mark44]

Yes.

You're making things harder than they need to be, though. Here's what you have:
sin⁴x - cos⁴x
= (sin² x - cos²x)(sin² x + cos²x)
= (sin² x - cos²x)
= -(cos² x - sin²x
= -cos 2x