Compare taxes Cobb-Douglass and more
The intuition you have is correct. Mathematically you can show it by first deriving the optimal choices with the lump sum income tax. So you will set up the following lagrangian:
$$\mathcal{L} = x^{1/2}_1 x^{1/2}_2 - \lambda [x_1p_1+x_2p_2 - m + T] $$
This gives you 3 FOC's the budget constraint and:
$$ 0.5x_1^{-0.5} x_2^{0.5} = \lambda p_1 \\
0.5x_2^{-0.5} x_2^{0.5} = \lambda p_2$$
Solve for optimal $x_1^*$ and $x_2^*$:
$$ x_1^* =  \frac{m-T}{2p_1} = \frac{32-T}{2} \\
x_2^* =  \frac{m-T}{2p_2} = \frac{32-T}{2}$$
Where here the second equalities take advantage of the assumptions that $p_1 = p_2=1$ and $m=32$.
Now  you can just plug this into the utility function and assuming you made no mistake equate this to the utility with the consumption tax on $p_1$ so you will have:
$$ 8 =  \left( \frac{32-T}{2}\right)^{0.5} \left( \frac{32-T}{2}\right)^{0.5} \\
T =16$$
So under the income tax regime government gets $T=16&gt; t=12$ while consumer still has the same utility as under the consumption tax, meaning the income tax is better. The intuition for that is that income tax does not distort the relative prices only has an income effect whereas consumption tax has both income and substitution effects.