I think I have this problem down but then again I'm not sure. If you could check my work, please:

cot^2x = 1-2cscx+csc^2x

csc^2x-1 = 1-2cscx+csc^2x (from the pythag. identity)

0 = 2-2cscx + csc^2x-csc^2x

0 = 2-2cscx

0 = -1+cscx (divided all by -2)

cscx = 1

Also, I am so stuck on the following two. I have to use the double (or half, but I'm sure it's double) angle formulas but I don't know how to properly incorporate them:

1. 2cos^2x-1 = cosx

and

2. cos2x + 6sin^2x = 2

Help would be very much appreciated. Thank you!