[TheDefiniteArticle]

Hi, I've been set a homework sheet and think I've made a mistake somewhere along this question:

Show that 1 + e^(jx) = (2cos((1/2)x))e^((1/2)jx)

My working thus far (may have mistakes in, I'm tired and braindead):

1 + e^(jx)
= 1 + cosx + jsinx
= 1 + (1/2)(2cos^2((1/2)x) + 1) + 2jsin((1/2)x)cos((1/2)x)
= (3/2) + cos^2((1/2)x) + 2jsin((1/2)x)cos((1/2)x)
= (3/2) + (2cos((1/2)x)(1/2cos((1/2)x) + jsin((1/2)x))

And naturally I would turn that last bit into e^((1/2)jx) if I had a whole cos((1/2)x), but it still wouldn't be the result I want.

Sorry if this is hard to read, but I'm hopeless with LaTeX.

Cheers.

[ttoby]

On this line:

(Original post by TheDefiniteArticle)
= 1 + (1/2)(2cos^2((1/2)x) + 1) + 2jsin((1/2)x)cos((1/2)x)

The 1/2 shouldn't be there, and the +1 should be -1.

You should find that things cancel/simplify from here.

[TheDefiniteArticle]

Well aren't I a daft ****.

Thanks.

[DPLSK]

(Original post by TheDefiniteArticle)
Show that .

I personally would have started with the RHS, using the following identity.

Aside: This can be shown using the fact that .

It should make life a lot easier.

I hope it helps.

Darren