If tan x = 1/2 and tan y = 1/3, Find the Value of x + y
Question:
If \[ \tan x=\frac12 \] and \[ \tan y=\frac13 \] then the value of \[ x+y \] is …………………………………….
If \[ \tan x=\frac12 \] and \[ \tan y=\frac13 \] then the value of \[ x+y \] is …………………………………….
Solution
Using the tangent addition formula:
\[ \tan(x+y) = \frac{ \tan x+\tan y } { 1-\tan x\tan y } \]
Substituting the given values,
\[ \tan(x+y) = \frac{ \frac12+\frac13 } { 1-\frac12\cdot\frac13 } \]
\[ = \frac{ \frac56 } { 1-\frac16 } \]
\[ = \frac{ \frac56 } { \frac56 } \]
\[ =1 \]
Therefore,
\[ \tan(x+y)=1 \]
Hence,
\[ x+y=\frac{\pi}{4} \]
Therefore,
\[ \boxed{\frac{\pi}{4}} \]