Compare taxes Cobb-Douglass and more Let a utility function for a consumer be defined as $u(x_{1},x_{2})=x_{1}^{1/2} x_{2}^{1/2}$. With the budget $x_{1}p_{1}+x_{2}p_{2}=m$. With values $p_1=p_2=1, m=32$. The state now adds a tax of unit 3 on $p_{1}$ (pr. Unit $x_1$)
How does it effect utility? What does the state earn?

I got the utility before to be 16 and after to be 8 with taxes correlating to 12 pr unit $x_1$

The state now considers an income tax such that the income is now $m-T$ How much will the state earn with the new system whilst keeping the consumer indifferent? Which system is better?

I figured that i solve for the Tax in the utility function under optimal demand conditions so that i kepy utility equal to 8. This gave me 16 units of income tax.

How does one do the last part mathematically. I figure the income tax is better for the consumer but how can i show it mathematically?
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.