40. The value of $x$ that satisfies $\cot ^{-1} 2=\cot ^{-1} x+\cot ^{-1} 7$ is ).
(A) 1
(B) 2
(C) 3
(D) 4
$(\mathrm{E}) *$
40. (C).

Let $\alpha=\cot ^{-1} 2 \cdot \beta=\cot ^{-1} 7$. Then $\tan \alpha=\frac{1}{2}, \tan \beta=\frac{1}{7}, \alpha, \beta \in\left(0, \frac{\pi}{4}\right)$.
$$
\begin{aligned}
\therefore \tan (\alpha-\beta) & =\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \cdot \tan \beta}=\frac{\frac{1}{2}-\frac{1}{7}}{1+\frac{1}{2}} \times \frac{1}{7} \\
& =\frac{1}{3} .
\end{aligned}
$$

Therefore, $\alpha-\beta=\cot ^{-1} 3$. Hence, $x=3$.