Proving Equivalence of certain Complex numbers

Hello Everyone! Today we are asked to prove that both this expressions are true: and . To be quite honest, I am not very sure as to how I can prove these, but my attempt for the first one, I was able to prove that but was unsure as to how I would be able to turn it into which is what I need to prove the first equality.

As for the second one I did something like this:

So that:

and by properties of odd and even function:

Thus, it should be true to say that :

The proof above, for me, is quite sloppy, but I think I could live with that. But for the first one, I really have no idea as to how to prove it. Anyone got some ideas as to how I should prove both equivalence? Thanks everyone in advance!(Wink)

Re: Proving Equivalence of certain Complex numbers

Surely you know that is NOT ...

Re: Proving Equivalence of certain Complex numbers

Quote:

Originally Posted by

**EliteAndoy** and

. To be quite honest, I am not very sure as to how I can prove these, but my attempt for the first one, I was able to prove that

but was unsure as to how I would be able to turn it into

which is what I need to prove the first equality.

You must know that and

Quote:

Originally Posted by

**EliteAndoy** .

As for the second one I did something like this:

So that:

and by properties of odd and even function:

Thus, it should be true to say that :

Yours above is the standard proof for this.

You might add that this works because cosine is an even function and sine is an odd function.

Re: Proving Equivalence of certain Complex numbers

Yeah, that's absolutely true,that's why I was unable to prove the equality. Here's how my little failed proof goes:

From the equation above, we can see that the modulus is 1 so that:

For the equation above to be true, the radicand should equal to 1, so that

From here, I tried to play with a lot of trig identities which all I was able to do is prove that this equation is true. But yeah, like Plato and Prove it said, this is not equal to

. So, anyone here have any idea as to how I should proceed with the proof? Thanks again to both of you!

Re: Proving Equivalence of certain Complex numbers

Quote:

Originally Posted by

**EliteAndoy** So, anyone here have any idea as to how I should proceed with the proof? Thanks again to both of you!

**What in the world are you on about?**

__ALL__ of those are __equals__. What don't you get?

Re: Proving Equivalence of certain Complex numbers

Damn it, I forgot about that. Well thanks for verifying that. :D

Re: Proving Equivalence of certain Complex numbers

Quote:

Originally Posted by

**EliteAndoy** (Swear) it, I forgot about that. Well thanks for verifying that. :D

Watch the language, please.

-Dan

Re: Proving Equivalence of certain Complex numbers

Oh sorry about that hehe.