(1) Because the included angle between B 1D and a1b is 30 degrees, A 1B 1=AB=2. It is known that A 1D=2√3/3 and A 1E= 1.

If we do EG⊥A 1B 1 in G, we can see that EG=√3/2, A 1E= 1/2.

With A 1A, A 1B 1 and A 1D as x, y and z axes, the spatial rectangular coordinate system A 1-xyz is established.

A 1(0，0，0)B 1(0，2，0)F( 1， 1，0)E(0， 1/2，√3/2)

A 1E=(0， 1/2，√3/2)，B 1F=( 1，- 1，0)

cos & ltA 1E，b 1F & gt； = -√2/4

(2)D(0，0，2√3/3)

A 1D=(0，0，2√3/3)B 1D=(0，-2，2√3/3)

The normal vector of face A 1B 1F is a1d = (0,0,2 √ 3/3), and let the normal vector of face DB 1F be n=(x, y, z).

At the same time n⊥DB 1, n⊥B 1F, so z=3 is n = (√ 3, √ 3).

cos & ltA 1D，n & gt=√ 15/5

The cosine of dihedral angle is √ 15/5, so the tangent of dihedral angle is √6/3 according to Pythagorean theorem.

The above is my answer according to the steps of the standard answer. Pure handwriting. Please give points. Thank you.