The value of $tan^{-1}5+tan^{-1}3-cot^{-1}\frac{4}{7}$ is

$\frac{\pi}{2}$

$\tan^{-1}5+\tan^{-1}3-\cot^{-1}\frac{4}{7}$

$\cot^{-1}\frac{4}{7}=\tan^{-1}\frac{7}{4}.$

$\tan^{-1}5+\tan^{-1}3=\pi-\tan^{-1}\frac{4}{7}\quad(\text{since }5\cdot3>1).$

$\tan^{-1}\frac{4}{7}+\tan^{-1}\frac{7}{4}=\frac{\pi}{2}.$

$\Rightarrow \tan^{-1}5+\tan^{-1}3-\tan^{-1}\frac{7}{4} =\pi-\frac{\pi}{2}.$

$=\frac{\pi}{2}.$

$\text{Required value }=\frac{\pi}{2}.$