SOLUTION: I've been given this problem and I keep getting stuck half was down it and going back on myself, wondered i f anyone could help out with it. Given that cos x = (e^jx + e^-jx

Question 90171: I've been given this problem and I keep getting stuck half was down it and going back on myself, wondered i f anyone could help out with it.
Given that
cos x = (e^jx + e^-jx)/2 and
sin x = (e^jx - e^-jx)/2j
and using only this information, prove that
cos^2 x - sin^2 x = 1 - 2sin^2 x
Answer by kev82(151) (Show Source): You can put this solution on YOUR website!
Hi

First thing to do is to tidy up the thing we're trying to prove. Reason being, we're going to have to a bit of messy algebra, and the less we have to do, the better. I'm going to add sin^2(x) to both sides. This leaves the thing we have to prove at:

Well, from here, all we do is substitute the definitions of sine and cosine in, then see what happens. To make it a bit simpler, I'm going to let

Using this, it's easy to see that

.

So using my new definitions,

and

Substitute these into the thing we are trying to prove, and you should get:

Multiply the whole lot by 4(we don't like denominators when we don't need them)

That just leaves exapnding the brackets, and tidying up

(Remember pq=1)

, so

. Which is always true, so what we started with must be true - proved.

Now try a different one yourself, see if only using your definitions of sine and cosine you can prove that

Kev
RELATED QUESTIONS
Hello! I'm stuck on this problem and think I'm doing it wrong. Any help? 3v-7+2v = 5v... (answered by rfer)

I am not sure if this is a quadratic actually, but I wasn't sure what selection in the... (answered by RAY100,drj,Theo)

You can afford a $1300 per month mortgage payment. You've found a 30-year loan at 7%... (answered by Theo)

I have been working on this problem and feel kind of stuck: --------------- "Consider (answered by Alan3354)

please help me solve this problem, I have tried to do it myself, but keep getting the... (answered by stanbon)

ok so im working with slopes and i just keep getting everything confused and im going a... (answered by Alan3354)

I am running out of time and I keep getting lost in this problem. Can anyone please... (answered by josgarithmetic)

how do graph this problem I tried to do it myself but keep getting it wrong Write... (answered by Fombitz)

I need help with this graphing problem I tried to do it myself but keep getting it wrong (answered by Mathtut)