The relationship between gross profit and tax
Most of what you have done was correct. Here are the steps to finish the argument. 

Evaluating the profits at the optimum, you get 
\begin{equation}
\pi_G=R(y^*(t))-C(y^*(t))
\quad\text{and}\quad 
\pi_N=R(y^*(t))-C(y^*(t))-ty^*(t).
\end{equation}
Differentiate the net profits with respect to $t$:
\begin{align}
\frac{\mathrm d\pi_N}{\mathrm dt}&amp;=R'(y^*)\frac{\mathrm dy^*}{\mathrm dt}-C'(y^*)\frac{\mathrm dy^*}{\mathrm dt}-t\frac{\mathrm dy^*}{\mathrm dt}-y^*\\&amp;=\underbrace{[R'(y^*)-C'(y^*)-t]}_{=0 \text{ due to FOC}}\frac{\mathrm dy^*}{\mathrm dt}-y^*&lt;0, 
\end{align}
which is true assuming $y^*&gt;0$. 
Differentiate the gross profits with respect to $t$: 
\begin{align}
\frac{\mathrm d\pi_G}{\mathrm dt}&amp;=R'(y^*)\frac{\mathrm dy^*}{\mathrm dt}-C'(y^*)\frac{\mathrm dy^*}{\mathrm dt}=[R'(y^*)-C'(y^*)]\frac{\mathrm dy^*}{\mathrm dt}&lt;0. 
\end{align}
This is true because from the FOC, we know that for any positive tax, $R'(y^*)-C'(y^*)&gt;0$, and since you've correctly derived $\frac{\mathrm dy^*}{\mathrm dt}&lt;0$, their product must be negative.