Question

If \(\tan^{-1}x + \tan^{-1}y + \tan^{-1}z = \pi\), then the value of \(x+y+z-xyz\) is

• A. 1
• B. 0
• C. -1
• D. 1/2

Hint:

Whenever sum of \(\tan^{-1}\) terms equals \(\pi\), directly use: \(x + y + z = xyz\).

Answer 1

The Correct Option is B

Concept: \[ \tan(A+B+C) = \frac{x+y+z - xyz}{1 - (xy+yz+zx)} \] where \(A = \tan^{-1}x\), \(B = \tan^{-1}y\), \(C = \tan^{-1}z\).

Step 1: Use given condition. \[ A+B+C = \pi \Rightarrow \tan(A+B+C) = \tan\pi = 0 \]

Step 2: Apply identity. \[ \frac{x+y+z - xyz}{1 - (xy+yz+zx)} = 0 \]

Step 3: Condition for fraction = 0. A fraction is zero when numerator = 0 and denominator \(\neq 0\): \[ x + y + z - xyz = 0 \]