# Simplifying with trig identities

• earthloop
In summary, the conversation discusses a simplification given in a textbook for an example question involving cos(pi+n*pi). The person is struggling to understand how the simplification was made and is seeking clarification on any missed tricks. The conversation also mentions the identity for separating out sine and cosine terms and applies it to the equation. It is pointed out that when n is an integer, sin(n*pi) equals 0. Ultimately, the person realizes their mistake and thanks the others for their help.
earthloop

## Homework Statement

[/B]
Hi, I am currently working through a textbook, and the following simplification is given for an example question:

I can't seem to work out how they have moved from cos(pi+n*pi) to cos(pi)cos(n*pi) so easily? Is there a simple trick I have missed? I understand the identity that separates out the sine and cosine terms (-cos(a+b) = sin(a)sin(b)-cos(a)cos(b)) but I'm having very little luck in getting the textbooks answer from that.

2. Homework Equations

cos(a+b) = cos(a)cos(b)-sin(a)sin(b)
cos(a-b) = cos(a)cos(b)+sin(a)sin(b)
-cos(a+b) = sin(a)sin(b)-cos(a)cos(b)
-cos(a-b) = -sin(a)sin(b)-cos(a)cos(b)

## The Attempt at a Solution

applying the identity to the cosine/sine part of the equation ONLY (ignoring 2/(1-n^2) ) where a = pi :

$-\frac{\left(n + 1\right)\, \left(\cos\!\left(a\, n\right)\, \cos\!\left(a\right) + \sin\!\left(a\, n\right)\, \sin\!\left(a\right)\right) + \left(n - 1\right)\, \left(\cos\!\left(a\, n\right)\, \cos\!\left(a\right) - \sin\!\left(a\, n\right)\, \sin\!\left(n\right)\right)}{n^2 - 1}$

Am I on the right track?

Thanks

Look at the cos(a+b) identity, what happens when you have sin(n*pi)?

n is assumed to be an integer here, correct?

jedishrfu... sorry not sure what you mean? Can you give a bit more detail?
Hallsofivy...yes n is an integer

oh! Thanks guys... Completely couldn't see the wood for the trees :)

sin(n*pi) = 0

Doh!

Cheers

Sometimes the wood is fossilized. :-)

## 1. What are trig identities and why do we use them?

Trig identities are mathematical equations involving trigonometric functions that are true for all values of the variables. We use them to simplify complex trigonometric expressions and solve equations.

## 2. How do I know which trig identity to use?

Choosing the right trig identity to use depends on the given expression and the goal of simplification. Familiarity with the commonly used identities and practice with solving problems will help you determine which identity to use.

## 3. Can I use multiple trig identities in one problem?

Yes, you can use multiple trig identities in one problem to simplify the expression further. It is important to use the correct identities in the correct order to avoid errors.

## 4. What are some common mistakes to avoid when using trig identities?

Some common mistakes to avoid when using trig identities include using the wrong identity, forgetting to apply the identity to the entire expression, and making sign errors. It is important to double-check your work and practice regularly to avoid these mistakes.

## 5. Are there any tips for simplifying with trig identities?

Yes, some tips for simplifying with trig identities include memorizing the commonly used identities, understanding the properties of trigonometric functions, and breaking down complex expressions into simpler ones. It is also helpful to practice solving problems regularly to improve your skills.

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