# What is an identity?

1. Jun 24, 2008

### LAVRANOS

can anybody give me the definition of a trig-identity?
And then the definition of an equation?
Because i think that the relation
$$\tan^2 x + 1 = \sec^2 x$$
is not an identity.

2. Jun 24, 2008

### dx

Re: proving trig identities

A trig identity is a proposition involving '=' and trigonometric functions that is true for all values of x. An equation is a proposition involving '=' which is not necessarily true for all x. $$\tan^2 x + 1 = \sec^2 x$$ is an identity because it is true for all x.

Last edited: Jun 24, 2008
3. Jun 24, 2008

### LAVRANOS

Re: proving trig identities

According to your definition when i say: The relation (sinx)^2+(cosx)^2 =1 is beautiful ,this is a trig identity.
Also when i put into my computer ,tan(90),tan270,tan450,tan630 e.t.c and i mean 90 ,270,450,630 deegres i get error which means that the trig (tanx)^2+1=(secx)^2 is not true for all the values of x

4. Jun 24, 2008

### rock.freak667

Re: proving trig identities

well in tan(90) is undefined

But it's actually true for all x.

take x=90

$tan(90)= \infty$ and sec(90)=1/cos(90)=$\infty$

so it is true for x=90 but not very useful.

5. Jun 24, 2008

### dx

Re: proving trig identities

Yes, that is a trig identity.

That's because tan is not defined at those values. When I said all values, I meant all values for which the expression is meaningful. For example, x + 0 = x is an identity. But if you put x = !"£%£" on your computer you get an error.

6. Jun 24, 2008

### LAVRANOS

Re: proving trig identities

But you said and i quote:an equation is aproposition involving = which is not necessarily true for all x.So you must decide is it an ID or an equation?
O.K rock.freak667 you mean that (infinity)^2 + 1= (infinity)^2 is true?
When i put into my computer x=90(degrees) and i get an error as an answer it means there is not such x to satisfy the trig equation

7. Jun 24, 2008

### dx

Re: proving trig identities

You get an error because tan x is not defined for x = 90, Just like 5/x is not defined for x = 0. But 5/x = 5/x is still an identity because its true for all values for which it is defined. By all values it is always meant all values for which the function is defined. In my definitions above, "all x" means "all meaningful x". You claim that $$\tan^2 x + 1 = \sec^2 x$$ is not an identity. By my definition of identity, it is. Whats your definition of identity?

8. Jun 24, 2008

### LAVRANOS

Re: proving trig identities

Allow me to go sleep now ,sorry for the delay but i was looking at darkfire"s 1=i thing .
Tomorrow we will Curry on

9. Jun 25, 2008

### tiny-tim

For some purposes, infinity is a number.

For trigonometric formulas, I personally regard the range as being R+ (the real numbers plus one point at infinity) rather than R.

It is silly to say that tan90º does not exist when R+ is a perfectly valid number system, and everyone know that tan90º = ∞.

So, for example, the formula tanx = sinx/cosx is an identity, and is valid even when cosx = 0.

10. Jun 25, 2008

### LAVRANOS

Re: proving trig identities

dx ,i asked you whether the sentance < The relation (sinx)^2 +(cosx)^2 =1 is beautiful > is atrig identity and you said yes .
Do still insist on that?
Also trhere is a great difference BETWEEN the words VALID and TRUE .VALID is an argument ,adeductuion alogical implication while TRUE OR FALSE IS a proposition or a sentance.
SO we can have a VALID argument with FALSE result.
HOW would you define an IDENTITY WITHIN logic(sympolic logic)????

11. Jun 25, 2008

### dx

Re: proving trig identities

No, I said (sinx)^2 +(cosx)^2 =1 is a trig identity. Statements like "I am beautiful" are not propositions unless there's a well defined meaning to "beautiful".

I don't know how one would precisely define identity. But everyone knows that things like (tanx)^2 + 1 = (secx)^2 are identities. If you don't agree, tell me what you think an identity is.

12. Jun 25, 2008

### Count Iblis

Re: proving trig identities

http://www.math.upenn.edu/~wilf/AeqB.html" [Broken] (section 1.5 is devoted to trig identities)

Last edited by a moderator: May 3, 2017
13. Jun 25, 2008

### CRGreathouse

Re: proving trig identities

Here's a shot at a basic definition of identity.

An identity is an an ordered triple $(A, E_1, E_2)$ where A is an algebraic structure and $E_1, E_2$ are expressions on A, where expressions are defined recursively as follows:
x is an expression on A
If $e_1,e_2,\ldots,e_n$ are expressions on A, and O is an n-ary operator on A, then $O(e_1,e_2,\ldots,e_n)$ is an expression on A. The identity is said to hold if and only if $E_1=E_2$ for all x in the underlying set of A.

A good definition would allow for:
* Undefined values outside the underlying set of A, where the identity holds iff both sides are defined and equal
* Parameterized unknowns

Last edited: Jun 25, 2008
14. Jun 25, 2008

### LAVRANOS

Re: proving trig identities

GRGreathouse where did you get that definition please tell me

15. Jun 25, 2008

### LAVRANOS

Re: proving trig identities

dx when your girlfriend tells that you are beautiful do you ask her to define beautiful?
You gonna loose her for ever

16. Jun 25, 2008

### dx

Re: proving trig identities

I'm not sure whether that's a joke or not. I never debate the meaning of "identity" with my girlfriend. Also, you're not my girlfriend, so I can ask you to define beautiful :)

Last edited: Jun 25, 2008
17. Jun 25, 2008

### LAVRANOS

Re: proving trig identities

You wonna bet that you cannot define beautiful??
Why i always ask my girlfriend for an ID JUST to make sure

18. Jun 25, 2008

### Defennder

Re: proving trig identities

This is getting irrelevant and personal. I'll just add in my 2 cents worth about the original question.

The trigo identities are an identity as far as mathematical theorems and proofs are concerned. a^2 + b^c = c^2, pythagoras theorem is also an identity as it can be proven.

Same goes for lots of other mathematical equations to name just a few random ones:

$$e^{i\pi} = -1$$
$$e^{i\theta} = \cos \theta - i \sin \theta$$
$$\vec{a} \cdot \vec{b} = |a||b| \cos \theta$$

All these are identities. So if they count as identities, why don't trigo identities count as identities?

19. Jun 25, 2008

### LAVRANOS

Re: proving trig identities

What is a mathematical equation.
Centairly a^2 +b^2 =c^2 is an ID,but what is e^iπ=-1?

20. Jun 25, 2008

### LAVRANOS

Re: proving trig identities

certainly,sorry