Ramanujan's tau function identity While studying Ramanujan's tau function, I observed that the function satisfies a beautiful identity that I had not seen previously in the literature.
Let $\tau(n)$ be Ramanujan's tau function, such that
$$q\prod_{n=1}^\infty (1-q^n)^{24} = \sum_{n=1}^\infty \tau(n)\,q^n,$$
then 
$$\tau(2)=8\dfrac{\tau(8)-2\tau(6)}{\tau(4)-15 \cdot 2^{11}}$$ 
I believe it is well known due to the popularity of the function. Can anyone prove the identity?
The identity implies the congruence property $$\tau(2)\equiv0\pmod{8}$$ 
Thus proving the identity is akin to proving the congruence property.

The identity follows immediately from the fact that 1728 can be expressed in terms of ramanujan tau function in two different ways,namely
$$2\tau(2){(\tau(3)-2^{11})}-\tau(8)=12^{3}$$
And 
$$-\dfrac{\tau(2){(\tau(4)+2^{11})}}{8}=12^{3}$$
After equating the two identities and using the multiplicative property of the function,
$\tau(mn)=\tau(m)\tau(n)$ for $\gcd(m,n)=1$,
Then we formally obtain  the identity after some simple algebraic manipulation.
