# Double Angle Formula

1. Dec 1, 2016

### Veronica_Oles

1. The problem statement, all variables and given/known data
Simplify cos^2 8x - sin^2x

2. Relevant equations

3. The attempt at a solution
I thought it would be in the format of cos2x
But I can't seem to figure it out I tried cos (4 * 2x)

And I tried to change the sin^2x into 1-cos^2x and I could get any farther.

Not sure how else to simplify.

2. Dec 1, 2016

### RUber

Try writing it as cos(2*4x), or let u = 4x and then do it with cos(2u).

EDIT:
Sorry, I rushed through reading your problem. What tools do you have other than the double angle formula? You might be able to write this as a difference of squares first, then apply some identities.

Do you know what the result should look like? How do you know when it is simple enough?

Thanks.

Last edited: Dec 1, 2016
3. Dec 1, 2016

### Ray Vickson

What, really, is meant by "simplify"?

The original result is about as simple as it gets. If you try to express everything in terms of $\cos(x)$ and $\sin(x)$ alone, your expression $\cos^2 (8x) - \sin^2 x$ becomes
$$1-\sin^2 x -64 \cos^2 x + 1344 \cos^4 x - 10752 \cos^6 x + 42240 \cos^8 x\\ - 90112 \cos^{10} x +106496 \cos^{12} x -65536 \cos^{14} x +16384 \cos^{16} x$$
Would you say that expression is simpler than the original one?

4. Dec 1, 2016

### Veronica_Oles

There is no solution unfortunately it was just a problem given:(

5. Dec 1, 2016

### Veronica_Oles

Yeah first one is definetly simpler.

6. Dec 2, 2016

### Buffu

I have done this problem before, In my book they wanted it to be
$\cos(9x)\cos(7x)$.

7. Dec 4, 2016

### lurflurf

use reduction identities
$$\cos^2(8x)=\frac{1+\cos(16x)}{2}\\ \sin^2(x)=\frac{1-\cos(2x)}{2}$$