Compute $\tan \left (\operatorname{arccot} \frac{4}{7} \right).$
I know that the arccotangent function is the inverse of the cotangent function, so if $\operatorname{arccot} \frac{4}{7} = x,$ then $\cot x = \frac{4}{7}.$ I also know that the tangent function is the reciprocal of the cotangent function, so $\tan x = \frac{1}{\cot x} = \frac{7}{4}.$ Therefore, $\tan \left (\operatorname{arccot} \frac{4}{7} \right) = \frac{7}{4}.$ 