trig identity for sqrt(2)/2

**rainer** · December 13th 2009, 05:26 AM

Well, obviously it's going to simplify in such a way as to equal sqrt(2)/2. That's sort of the whole point of an identity isn't it.

The challenge was to come up with a trig identity that equals this and that is valid for all angles without using the number two. A couple days ago you said such a thing didn't exist. I've demonstrated that it does exist. That seems to me to constitute something new.

**Wilmer** · December 13th 2009, 07:47 AM

Ok. Why ask us?

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**Defunkt** · December 13th 2009, 10:01 AM

The whole point of an identity is that it's not derived strictly from other identities. Take for example the pythagorean identity $sin^2\theta + cos^2\theta = 1$ . As far as I'm concerned, it can not be derived from any algebraic derivation of another identity (which is exactly what your "identity" is...)

Now,
$\frac{1}{1 + 2sin^2\theta + cos(2\theta)} \equiv \frac{1}{\sqrt{2}} \forall \theta$ , and I might as well come up with 50 other "identities", all equal $\frac{1}{\sqrt{2}}$ , which are all based upon the pythagorean identity.

If I saw $\frac{1}{\sqrt{\cos^2\theta(\tan^2\theta+\sec^2\th eta+1)}}$ somewhere, I would be able to simplify it to $\frac{1}{\sqrt{2}}$ . But can you say that you will know how to show $sin^2\theta + cos^2\theta = 1$ ?

Do you understand why identities are called as such?

**rainer** · December 14th 2009, 04:42 AM

Defunkt,

Before seeing your reply I was thinking this over and realized you are right. What I have come up with is a tautology, not an identity.

It does still work as a puzzle, though. Another good puzzle would be to express the same thing with hyperbolic functions without using the number 2.

As someone trying to understand math without a professor and without books, I must often rely on folly as my teacher. At the very least I hope my folly has provided some comic relief for everyone.

**Wilmer** · December 14th 2009, 07:39 AM

Originally Posted by rainer

At the very least I hope my folly has provided some comic relief for everyone.

I see nothing "comically wrong" with your "buts, ifs..."; shows you have
an inquisitive mind; also quite a good way to learn "by default"

**Drexel28** · January 9th 2010, 04:31 PM

Originally Posted by Defunkt

Take for example the pythagorean identity $sin^2\theta + cos^2\theta = 1$ . As far as I'm concerned, it can not be derived from any algebraic derivation of another identity (which is exactly what your "identity" is...)

Really?

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