# Proving trig identities with euler's

• schapman22
In summary, the conversation discusses using Euler's identity to prove two trigonometric identities. The conversation includes a step-by-step attempt at solving the problem and a question about a potential typo in the provided instructions. The expert confirms the typo and provides the correct equation to use.
schapman22

## Homework Statement

Use Euler's identity to prove that cos(u)cos(v)=(1/2)[cos(u-v)+cos(u+v)]
and sin(u)cos(v)=(1/2)[sin(u+v)+sin(u-v)]

## Homework Equations

eui=cos(u) + isin(u)
e-ui=cos(u)-isin(u)

## The Attempt at a Solution

I was able to this with other trig identities with no problem but this one I have hit a wall.
cos(u+v)+isin(u+v)+cos(u-v)+isin(u-v)=eu(cos(v)+isin(v)+cos(v)-isin(v)) then
equating the real parts
cos(u+v)+cos(u-v)=eu(2cos(v)) then divide by 2
(1/2)[cos(u+v)+cos(u-v)]=eu(cos(v))

I cannot figure out why I have an eu and not a cos(u). Does anyone see where I have gone wrong or what I am missing? Thank you in advance.

hi schapman22!
schapman22 said:

you're missing an "i"

e(u+v)i+e(u-v)i=eui(evi+e-vi)

I'm looking at my worksheet and it says to use eu(evi+e-vi)
Are you certain of that. It could be a typo because my teacher hand writes all of our assignments.

schapman22 said:
Are you certain of that.

yup!

look at it!

Thanks, I really wish my teacher would use the book. This is like the 5th time I've spent hours on a problem only to find out there's a typo in it haha. I appreciate it.

## 1. What is Euler's identity and how is it used to prove trigonometric identities?

Euler's identity is a mathematical equation that states: eix = cos(x) + i sin(x). This equation relates the exponential function, trigonometric functions, and the imaginary unit. It can be used to prove trigonometric identities by converting trigonometric functions into exponential functions and then using Euler's identity to simplify the equation.

## 2. Can you provide an example of how Euler's identity is used to prove a trigonometric identity?

One example is proving the identity: cos(x) = (eix + e-ix)/2. Using Euler's identity, we can rewrite the right side of the equation as (cos(x) + i sin(x) + cos(x) - i sin(x))/2. Simplifying this, we get cos(x), which is equal to the left side of the identity.

## 3. Are there any limitations to using Euler's identity to prove trigonometric identities?

Yes, there are certain identities that cannot be proven using Euler's identity alone. For example, identities involving hyperbolic functions or inverse trigonometric functions cannot be proven using this method.

## 4. Is it necessary to have a deep understanding of Euler's identity to prove trigonometric identities?

While having a thorough understanding of Euler's identity can be helpful, it is not necessary to prove trigonometric identities using this method. Knowing how to manipulate equations and use basic algebraic principles is generally sufficient.

## 5. Can Euler's identity be used to prove identities involving multiple trigonometric functions?

Yes, Euler's identity can be used to prove identities involving multiple trigonometric functions. By converting the trigonometric functions into exponential form and using algebraic manipulations, multiple functions can be simplified into one function, ultimately proving the identity.

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