Thread: trig identity for sqrt(2)/2

1. 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.

2. Ok. Why ask us?

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3. The whole point of an identity is that it's not derived strictly from other identities. Take for example the pythagorean identity $\displaystyle sin^2\theta + cos^2\theta = 1$. 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,
$\displaystyle \frac{1}{1 + 2sin^2\theta + cos(2\theta)} \equiv \frac{1}{\sqrt{2}} \forall \theta$, and I might as well come up with 50 other "identities", all equal $\displaystyle \frac{1}{\sqrt{2}}$, which are all based upon the pythagorean identity.

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

Do you understand why identities are called as such?

4. 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.

5. 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"

6. Originally Posted by Defunkt
Take for example the pythagorean identity $\displaystyle sin^2\theta + cos^2\theta = 1$. 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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