The sum of the 6th power identities, huh?. Well, here is my way. Years ago

when I figured this out, I thought I had done something. But, it would be silly

to think it was original, afterall.

Consider

Notice the coefficients?. They look binomial, don't they?

Now, plug in the respective identities for the powers. We must know them dong it this way.

Now, solve for

It is lotsa horrible algebra, but if done correctly you should get:

I am sure there is a simpler way, but you can do any of them this way as long as you know the previous ones. Just plod through the algebra.