I really like to use paradoxes in my math classes, in order to awaken the interest of my students. Concretely, these last weeks I am proposing paradoxes that achieve the conclusion that **2=1**. After one week, I explain the solution in the blackboard and I propose a new one. For example, I posted the following one some months ago: What is wrong with the sum of these two series?
I would like to increase my repertoire of fake-proofs. I would be glad to read your proposals and discuss them! My students are 18 years old, so don't be too cruel :) Here is my own contribution:

\begin{equation} y(x) = \tan x \end{equation} \begin{equation} y^{\prime} = \frac{1}{\cos^{2} x} \end{equation} \begin{equation} y^{\prime \prime} = \frac{2 \sin x}{\cos^{3} x} \end{equation} This can be rewritten as: \begin{equation} y^{\prime \prime} = \frac{2 \sin x}{\cos^{3} x} = \frac{2 \sin x}{\cos x \cdot \cos^{2} x} = 2 \tan x \cdot \frac{1}{\cos^{2} x} = 2yy^{\prime} = \left( y^{2} \right)^{\prime} \end{equation} Integrating both sides of the equation $y^{\prime \prime} = \left( y^{2} \right)^{\prime}$: \begin{equation} y^{\prime} = y^{2} \end{equation} And therefore \begin{equation} \frac{1}{\cos^{2} x} = \tan^{2} x \end{equation} Now, evalueting this equation at $x = \pi / 4$ \begin{equation} \frac{1}{(\sqrt{2}/2)^{2}} = 1^{2} \end{equation} \begin{equation} 2 = 1 \end{equation}