can you prove these are equivalent without a calculator?

You can approach this two ways: In the easier one, take the two sides of the equation and square them. Soon you should see you can reorder the lhs (left hand side) to look like the rhs.

((sqrt(2) - sqrt(6))/4 )^2 = (2 - 2*sqrt(2)*sqrt(6) + 6 )/16 = (8 - 4*sqrt(3))/16 = (2 - sqrt(3)) / 4 (lhs)

(-sqrt(2-sqrt(3))/2 )^2 = (2 - sqrt(3)) /4 (rhs)

The other way is to start with the rhs and multiply the numerator and denominator with 2. Then take the 2 inside the sqrt as 4. The trick is to recognize the argument of the sqrt as a square at that point, which is not very trivial.

(√2 - √6) / 4 =? -√(2 - √3) / 2

square both sides.

(2 - 2√2√6 + 6) / 16 =? (2 - √3)/4

(8 - 4√3) / 16 =? (2 - √3)/4

(2 - √3) / 4 =? (2 - √3)/4

okay, so the squares are equal, but are the expressions equal?

√6 > √2, so (√2 - √6) / 4 < 0

2 > √3, so (2 - √3) > 0 and -√(2 - √3) <0

the expressions are both negative and their squares are equal

therefore they are equal.