(1)

3角形関数は偶関数なので明らかに成り立つ。

(2)

\begin{align*} \mathrm{rect}\left(x\right)*_{x}\mathrm{rect}\left(x\right) & =\int_{-\infty}^{\infty}\mathrm{rect}\left(t\right)\mathrm{rect}\left(x-t\right)dt\\ & =\int_{-\frac{1}{2}}^{\frac{\text{1}}{2}}\mathrm{rect}\left(t\right)\mathrm{rect}\left(x-t\right)dt\\ & =\int_{-\frac{1}{2}}^{\frac{\text{1}}{2}}\mathrm{rect}\left(x-t\right)dt\\ & =\int_{-\frac{1}{2}}^{\frac{\text{1}}{2}}\mathrm{rect}\left(t-x\right)dt\\ & =\int_{-\frac{1}{2}-x}^{\frac{\text{1}}{2}-x}\mathrm{rect}\left(t\right)dt\cmt{t-x\rightarrow t}\\ & =\int_{-\frac{1}{2}-x}^{\frac{\text{1}}{2}-x}\left(H_{\frac{1}{2}}\left(t+\frac{1}{2}\right)-H_{\frac{1}{2}}\left(t-\frac{1}{2}\right)\right)dt\\ & =\left[\left(t+\frac{1}{2}\right)H_{\frac{1}{2}}\left(t+\frac{1}{2}\right)-\left(t-\frac{1}{2}\right)H_{\frac{1}{2}}\left(t-\frac{1}{2}\right)\right]_{-\frac{\text{1}}{2}-x}^{\frac{\text{1}}{2}-x}\\ & =\left(1-x\right)H_{\frac{1}{2}}\left(1-x\right)-\left(-x\right)H_{\frac{1}{2}}\left(-x\right)-\left\{ \left(-x\right)H_{\frac{1}{2}}\left(-x\right)-\left(-1-x\right)H_{\frac{1}{2}}\left(-1-x\right)\right\} \\ & =\left(1-x\right)H_{\frac{1}{2}}\left(1-x\right)+2xH_{\frac{1}{2}}\left(-x\right)-\left(1+x\right)H_{\frac{1}{2}}\left(-1-x\right)\\ & =\begin{cases} 1-x+2xH-\left(1+x\right) & x<-1\\ 1-x+2x & -1\leq x<0\\ 1-x & 0\leq x<1\\ 0 & 1\leq x \end{cases}\\ & =\begin{cases} 0 & x<-1\\ 1+x & -1\leq x<0\\ 1-x & 0\leq x<1\\ 0 & 1\leq x \end{cases}\\ & =\mathrm{tri}\left(x\right) \end{align*}