Question

If $\tan^{-1}x + \tan^{-1}y + \tan^{-1}z = \pi$, then $x + y + z$ is

• A. $xyz$
• B. 0
• C. 1
• D. $2xyz$

Hint:

Key identity: if $\tan^{-1}x + \tan^{-1}y + \tan^{-1}z = \pi$, then $x + y + z = xyz$. This is a standard result for inverse trigonometry.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Use the inverse tangent addition formula and the given sum to derive a relation between $x$, $y$, and $z$.
Step 2: Detailed Explanation:
$\tan^{-1}x + \tan^{-1}y = \pi - \tan^{-1}z$.
Taking $\tan$ of both sides: $\dfrac{x+y}{1-xy} = \tan(\pi - \tan^{-1}z) = -z$.
$\Rightarrow x + y = -z(1-xy) = -z + xyz \Rightarrow x + y + z = xyz$.
Step 3: Final Answer:
$x + y + z = xyz$.