Question
If
\[ \tan^{-1}x + \tan^{-1}y = \frac{\pi}{4} \]
Find the relation between \( x \) and \( y \).
Solution
Take tangent on both sides:
\[ \tan\left(\tan^{-1}x + \tan^{-1}y\right) = \tan\frac{\pi}{4} \]
Using identity:
\[ \tan(A+B) = \frac{\tan A + \tan B}{1 – \tan A \tan B} \]
So,
\[ \frac{x + y}{1 – xy} = 1 \]
Cross multiply:
\[ x + y = 1 – xy \]
Final Answer:
\[ \boxed{x + y = 1 – xy} \]
Key Concept
Use tangent addition formula when inverse tangent expressions are added.