De Moivre's Theorum and Double-Angle Formulas

[Jessehk]

I hope this is in the right place.

I'm in grade 12, and I've been given an assignment involving complex numbers.

The question reads:

Use De Moivre's Theorum to verify the identities:
$$cos(2\theta) = cos^2\theta - sin^2\theta$$

$$sin(2\theta) = 2sin\theta cos\theta$$

I've tried something like this:
$$cos(2\theta) + i \cdot sin(2\theta) = (cos\theta + i \cdot sin\theta)^2$$

$$cos(2\theta) + i \cdot sin(2\theta) = cos^2\theta + i \cdot 2cos\theta sin\theta - sin^2\theta$$

$$cos(2\theta) = cos^2\theta - sin^2\theta + i \cdot 2cos\theta sin\theta - i \cdot sin(2\theta)$$

But I don't understand where to go from there. Can I somehow "separate" them?
Any help would be appreciated.

 

[Rainbow Child]

When the two complex numbers

$$a+i\,b, \quad c+i\,d$$

are equal?

 

[Jessehk]

When the two complex numbers

$$a+i\,b, \quad c+i\,d$$

are equal?

I'm sorry, I don't understand.

 

[Rainbow Child]

The equation $$a+i\,b=c+i\,d$$ gives

$$a=c,\, \quad b=d$$.

Apply this to your formulas

 

[Jessehk]

The equation $$a+i\,b=c+i\,d$$ gives

$$a=c,\, \quad b=d$$.

Apply this to your formulas

Well, I didn't know that.
Thanks for the help. :)

EDIT: I just got it: I'm an idiot. Thanks again.