If x(t) ↔ X(f) denotes a Fourier transform (FT) pair, $\mathrm{\Pi }\left({\displaystyle \frac{t}{T}}\right)$ denotes a rectangular pulse of width T and '⋆' denotes the convolution operation, then the FT of the signal $x(t)=\mathrm{\Pi }\left({\displaystyle \frac{t}{T}}\right)\ast \mathrm{\Pi }\left({\displaystyle \frac{t}{T}}\right)$ is

Concept:

1. Fourier transform of a function is defined as:

$X\left(\omega \right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}x\left(t\right)e^{-j\omega t}dt$

$X\left(f\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}x\left(t\right)e^{-j2\pi ft}dt$

2. Inverse Fourier transform is defined as:

$x\left(t\right)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}X\left(\omega \right)j^{j\omega t}d\omega$

$x\left(t\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}X\left(f\right)e^{j2\pi ft}df$

$As\pi (\frac{t}{\tau })=rect(\frac{t}{\tau })$

3. Fourier Transform of Rectangular pulse is:

Arect(t/T) = ATsinc(fT)  

4. Fourier Transform of Triangular pulse:

Atri(t/T) = ATsinc²(fT)   

5. Convolution in time domain = Multiplication in frequency domain

Calculation : 

Let x(t) is $Arect\left(\frac{t}{T}\right)$  

$Arect\left(\frac{t}{T}\right)=A\cdot G_T\left(t\right)$ and represented as:

$x\left(t\right)=u\left(t+\frac{T}{2}\right)-u\left(t-\frac{T}{2}\right)$

With shifting property, we get the Fourier transform as:

$X\left(\omega \right)=\frac{1}{j\omega }\left[e^{j\omega \frac{T}{2}}-e^{-j\omega \frac{T}{2}}\right]$

$X\left(\omega \right)=\frac{2}{j\omega \times 2}\left[e^{j\omega \frac{T}{2}}-e^{-j\omega \frac{T}{2}}\right]$

$=\frac{2}{\omega }\sin \left(\omega \frac{T}{2}\right)$

Let:

$S_a\left(x\right)=\frac{\sin \left(x\right)}{x}$

The sinc function is defined as:

$\sin cx=\frac{\sin \pi x}{\pi x}$

$X\left(\omega \right)=\frac{2\sin \left(\omega \frac{T}{2}\right)}{\omega \frac{T}{2}\times \frac{2}{T}}=T\cdot S_a\left(\frac{\omega T}{2}\right)$

$X\left(\omega \right)=T\cdot \sin c\left(\frac{\omega T}{2\pi }\right)$

$X\left(\omega \right)=T\sin c\left(\frac{\omega T}{2\pi }\right)$

$X\left(\omega \right)=T\sin c(fT)$

Now, the Multiplication in one domain corresponds to the convolution in another domain

$\mathrm{\Pi }\left({\displaystyle \frac{t}{T}}\right)\ast \mathrm{\Pi }\left({\displaystyle \frac{t}{T}}\right)=T^{2}sin{c}^2(fT)$

Hence option(1) is the correct answer.