Can anybody provide me with any information about this fascinating equation?
$$2^2 = \left(x+\frac{1}{x}\right)^2 - \left(x-\frac{1}{x}\right)^2 $$
I have been told many different things:
1) It is a mathematical Identity.
2) It is not a mathematical identity (because it does not work for zero).
3) It is a special case of the difference of squares equation.
Can anybody give me any concrete facts? The reason i ask is because it's so interesting. For example:
We can link this equation with Pythagoras’s theorem, so that for each number (x) we have a corresponding right angled
triangle and each triangle has height (A) = 2, base (B) = x-1/x and hypotenuse (C) = x+1/x.
As you can see, we can also link it with Trigonometry.
And because tangents to an arc are always reciprocals, we can use it to get the following equations:
SO, why does an equation this interesting not have a name (or does it?) and why is it not more well known?
I have been playing around with the geometry of this equation for years and it has allowed me to create many tools to help visualize the symmetry of a number or angle, for example:
We can also link this equation with the metallic means.
Does this equation work for zero?
Assuming that (x) is always >= 1 and that (1/x) is always <= 1.
As you can see, these variables always cancel each other out. Substituting 1/x = 0 (zero) and x = ∞ (infinity).
Tangents to an arc are always reciprocals.
The identity $$4xy=(x+y)^2-(x-y)^2$$ can be used to show that if two numbers have a fixed sum, their product is greatest when they are equal, as well as the AM/GM inequality for two numbers. It is well known and useful.
If we fix $xy=C$ then the identity $4C=(x+\frac Cx)^2-(x-\frac Cx)^2$ applies where all the terms are defined ie for $x\neq 0$ (note that $C$ need not be positive here).