if the LHS of an identity is symmetric does it mean the RHS must also be symmetric?
In addition how do you test if an identity in three variables is symmetric
e.g let the three variables be x,y and z
do you replace x with y, y with z and z with x - if it is the same equation then the equation is symmetric - is this a valid test?
If you talk about identities then you are right. If you have an identity $$F(x,y,z)=G(x,y,z)$$ and the LHS is symmetric in its variables, $$F(x,y,z)=F(x,z,y)=F(y,x,z)=F(y,z,x)=F(z,x,y)=F(z,y,x)$$ then the RHS is also symmetric.
If you talk about equations then you can't expect symmetry in its solutions. Think of $x^2+y^2=x.$ The LHS is a symmetric expression in $x,y$ while the RHS is not symmetric. $(1,0)$ is a solution while $(0,1)$ is not.
Yes.
Every identity is equivalent to a claim that two functions are equal. For example, the following is a (valid) identity from trigonometry.
$$(\forall \theta,\theta' : \mathbb{R})(\sin(\theta+\theta') = \sin \theta \cos \theta+\cos \theta \sin \theta')$$
(You can read about the meaning of $\forall$ here.)
But this is just saying that two functions are equal:
$$(\lambda \theta,\theta' : \mathbb{R})(\sin(\theta+\theta')) = (\lambda \theta,\theta' : \mathbb{R})(\sin \theta \cos \theta'+\cos \theta \sin \theta')$$
(You can read about the meaning of $\lambda$ here, although the linked page overcomplicates things.)
Of course, if two functions are equal, then they have exactly the same properties. For example, since the function $(\lambda \theta,\theta' : \mathbb{R})(\sin(\theta+\theta'))$ is symmetric in its two arguments, hence the function $(\lambda \theta,\theta' : \mathbb{R})(\sin \theta \cos \theta'+\cos \theta \sin \theta')$ is also symmetric in its two arguments.
