Confusing Symbol - Plane Trigonometry Book I am currently reviewing a book titled "A Treatise in Plane Trigonometry" by Hobson that was written in $1891$ and it does not  always explain use of symbols.
The author shows two trig identities (I want to prove these)
$$\tag 1 \sum \sin 2A \sin^2 (B+C) - \sin 2A \sin 2 B \sin 2C = 2 \sin(B+C) \sin(C+A) \sin(A+B) \\  \sum \cos 2A \cos^2 (B+C) - \cos 2A \cos 2 B \cos 2C = 2 \cos(B+C) \cos(C+A) \cos(A+B)$$
He then states that these two identities are based on the  algebraic identity
$$\tag 2 \sum 2a (b+c)^2 -8abc = 2(b+c)(c+a)(a+b)$$
In both cases, I do not understand how he is using the $\sum$ symbol. Is it only applying to the first item on the LHS or both terms? 
I have tried a few permutations on $(2)$, but am still not seeing it and it is probably something simple.
Can anyone shed light on how to properly read and derive $(2)$?
 A: I believe that $\sum$ is used for circular sum, that is: $$\sum2a(b+c)^2=2a(b+c)^2+2b(a+c)^2+2c(a+b)^2.$$
At least, the claimed algebraic identity is true using this interpretation.
A: 
In both cases, I do not understand how he is using the $\sum$ symbol. Is it only applying to the first item on the LHS or both terms? 

First item, only, $\sum x + y$ is usually understood to mean $(\sum x) + y \ne \sum (x+y)\,$.

$$\sum 2a (b+c)^2 -8abc = 2(b+c)(c+a)(a+b)$$

The LHS is a cyclical sum, often written as $\sum_{cyc}\,$:
$$\sum_{cyc} 2a (b+c)^2 -8abc = 2a(b+c)^2+2b(c+a)^2+2c(a+b)^2 - 8abc$$

Can anyone shed light on how to [...] derive (2) ?

You can simply expand, collect and cancel terms on both sides. Or, for a shortcut, note that the LHS is $0$ for $a=-b$, so (considered as a polynomial in $a$) it must have a factor of $(a+b)$. By symmetry, it must also have factors of $(b+c)$ and $(c+a)\,$, then what's left must be a constant (by degree considerations). Finally, take for example $a=b=c=1$ to figure out the constant factor of $2$.
