# Identities v Laws

I have tried to find this out through google and searching this site with no luck. Basically, are the terms 'law' and 'identity' interchangeable in Mathematics? What is described as 'logarithmic identities' in one place is referred to as 'log laws' elsewhere. Similiarly, 'index laws' and 'exponential identities'.

• Laws, sometimes, are meant to be axioms and/or rules, so identities and laws aren't interchangeable. – Git Gud Nov 17 '16 at 0:41
• I'm not sure, but I think that an "identity" is an equation of two functions, that is, a statement that two functions are equal. So, an identity is a law, but not all laws are identities. – Ben Grossmann Nov 17 '16 at 1:04
• 'Law' that can't be replaced by 'identity': law of large numbers, law of non-contradiction, ... – user251257 Nov 17 '16 at 1:11
• Note: two functions that are equal are said to be "identical" or "identically equal" – Ben Grossmann Nov 17 '16 at 1:22

## 1 Answer

In trigonometry, there are identities such as $\cos^2(x)+\sin^2(x)=1$ and $\sin(2x)=2\sin(x)\cos(x)$. But then there are "laws" such as $c^2=a^2+b^2-2ab\cos(c)$. The law of sines and law of cosines are not identities because they are true only for triangles in the Euclidean plane. There are different but analogous laws for triangles in the hyperbolic and elliptic plane.

So in trigonometry there is a distinction between an identity and a law.

• So Laws are more generalized/universal than identities? I guess that makes sense, but still wondering why 'index laws' and 'exponential identities' seem to describe the same thing? Seems to be something to do with the whole "all laws are identities but not all identities are laws" idea alluded to above, but struggling to get my head around it. – user5421817 Nov 17 '16 at 2:18