Simplifying Double Angle Formula: Cos^2 8x - Sin^2x

[Veronica_Oles]

Homework Statement

Simplify cos^2 8x - sin^2x

Homework Equations

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.

 

[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.

 

[Ray Vickson]

Homework Statement

Simplify cos^2 8x - sin^2x

Homework Equations

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.

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?

 

[Veronica_Oles]

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.

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

 

[Veronica_Oles]

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?

Yeah first one is definately simpler.

 

[Buffu]

Yeah first one is definitely simpler.

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

 

[lurflurf]

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