Question

If $\tan x=\frac{1}{2}$ and $\tan y=\frac{1}{3},$ then the value of x + y is _____________.

Answer 1

$$\begin{gathered} \mathrm{Given} \\ \tan x=\frac{1}{2} \\ \tan y=\frac{1}{3} \end{gathered}$$
$\begin{array}{rcl}\mathrm{Since}\tan \left(x+y\right)& =& \frac{\tan x+\tan y}{1-\tan x\tan y}\\ & =& \frac{{\displaystyle \frac{1}{2}+\frac{1}{3}}}{1-{\displaystyle \frac{1}{2}}\times {\displaystyle \frac{1}{3}}}\\ & =& \frac{{\displaystyle \frac{3+2}{6}}}{{\displaystyle \frac{6-1}{6}}}\\ & =& \frac{5}{6}\times \frac{6}{5}\end{array}$
$$\begin{gathered} \tan \left(x+y\right)=1 \\ \mathrm{i}.\mathrm{e}.x+y={\tan }^{-1} \\ \mathrm{hence},\mathrm{value}\mathrm{of}x+y\mathrm{is}\frac{\mathrm{\pi }}{4} \end{gathered}$$