When a body suspended by the string situated at position $P$ as shown in the figure, where latitude is $\lambda$ , then body is also rotated with angular frequency $\left(\omega \right)$ of earth, hence tension on the string is given by

$T=mg-m{\omega }^2\mathrm{c}\mathrm{o}\mathrm{s}\lambda$
$T=m.\frac{GM}{R^2}-m{\omega }^2\mathrm{c}\mathrm{o}\mathrm{s}\lambda \left[\therefore g=\frac{GM}{R^2}\right]$
$T=\frac{GMm}{R^2}-mr{\omega }^2\mathrm{c}\mathrm{o}\mathrm{s}\lambda$ …. (i)
When body is suspended at equator, then
$\lambda =0$ and $r=R$
$\therefore$ From Eq. (i), we have,
$T=\frac{GMm}{R^2}-mR{\omega }^2\mathrm{c}\mathrm{o}\mathrm{s}{0}^o$
$T=\frac{GMm}{R^2}-mR{\omega }^2$