Derive an expression for the potential energy of an electric dipole placed in a uniform electric field?

Consider an electric dipole placed in a uniform electric field $\overrightarrow{\mathrm{E}}$ with its dipole moment $\overrightarrow{\mathrm{P}}$ makes an angle $\mathrm{\theta }$ with field. Two forces on two charges creates couple will be

$\mathrm{\tau }=\mathrm{qE}\times 2\mathrm{asin}\mathrm{\theta }=\mathrm{PEsin}\mathrm{\theta }$

If the dipole is rotated through a small angle d$\mathrm{\theta }$ against the torque acting on it, then small work done is $\mathrm{w}=\mathrm{\tau d\theta }$
$\mathrm{dw}=\mathrm{PEsin\theta d\theta }$
The total work done in rotating the dipole from angle ${\mathrm{\theta }}_{1 }\mathrm{to} {\theta }_2$ with the direction of field
$\begin{array}{l}\mathrm{w}=\int \mathrm{dw}={\int }_{{\mathrm{\theta }}_1}^{{\mathrm{\theta }}_2} \mathrm{PEsin\theta d\theta }\\ =\mathrm{PE}{\int }_{{\mathrm{\theta }}_1}^{{\mathrm{\theta }}_2} \mathrm{sin\theta d\theta }=\mathrm{PE}[-\mathrm{cos\theta }{]}_{{\mathrm{\theta }}_1}^{{\mathrm{\theta }}_2}\end{array}$
$\mathrm{w}=\mathrm{PE}\left(\cos {\mathrm{\theta }}_1-\cos {\mathrm{\theta }}_2\right)$
This work done is stored as potential energy of dipole
$\mathrm{u}=\mathrm{PE}\left[{\mathrm{cos\theta }}_1-{\mathrm{cos\theta }}_2\right]$