Provee the identity (4-i*square root6)/(2-i*square root6)=(square root3+2i*square root2)/(square root3+i*square root2)

We have to prove that (4 - i*sqrt 6)/(2 - i*sqrt 6)=(sqrt
3 + 2i*sqrt 2)/(sqrt 3 + i*sqrt 2)

The left hand
side:

(4 - i*sqrt 6)/(2 - i*sqrt
6)

multiply the numerator and denominator by (2 + i*sqrt
6)

=> (4 - i*sqrt 6)*(2 + i*sqrt 6)/(2 - i*sqrt
6)*(2 + i*sqrt 6)

=> (8 + 4*i*sqtr 6 - 2*i*sqrt 6 +
6)/(4 + 6)

=> (14 + 2*i*sqtr
6)/10

=> (7 + i*sqrt
6)/5...(1)

The right hand
side:

(sqrt 3 + 2i*sqrt 2)/(sqrt 3 + i*sqrt
2)

multiply the numerator and denominator by (sqrt 3 -
i*sqrt 2)

=> (sqrt 3 + 2i*sqrt 2)*(sqrt 3 - i*sqrt
2)/(sqrt 3 + i*sqrt 2)*(sqrt 3 - i*sqrt 2)

=> (3 +
2i*sqrt 6 - i*sqrt 6 + 4) / (3 + 2)

=> (7 + i*sqrt
6)/5 ...(2)

As (1) and (2) are the same the identity is
proved.

The identity is proved
as both the sides are equal to (7 + i*sqrt
6)/5.