Question

If tan⁻¹ 3 + tan⁻¹ x = tan⁻¹ 8, then x =
(a) 5
(b) 1/5
(c) 5/14
(d) 14/5

Answer 1

(b) $\frac{1}{5}$
We know that ${\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\frac{x+y}{1-xy}$.
Now,
$$\begin{gathered} {\tan }^{-1}3+{\tan }^{-1}x={\tan }^{-1}8 \\ \Rightarrow {\tan }^{-1}\left(\frac{3+x}{1-3x}\right)={\tan }^{-1}8 \\ \Rightarrow \frac{3+x}{1-3x}=8 \\ \Rightarrow 3+x=8-24x \\ \Rightarrow 3-8=-24x-x \\ \Rightarrow -5=-25x \\ \Rightarrow x=\frac{5}{25}=\frac{1}{5} \end{gathered}$$