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Sum of three tan^-1 equal to pi or pi/2: relation among the numbers

This type of question, asked 2 times: 10 May 2024 S2, 20 Apr 2025 S1. All 2 questions below, oldest first.

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  1. Question 1

    10 May 2024 S2

    127
    If tan⁡−1a+tan⁡−1b+tan⁡−1c=π\tan^{-1}a+\tan^{-1}b+\tan^{-1}c=\pi, then which of the following statement is true?

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  2. Question 2

    20 Apr 2025 S1

    140
    If tan⁡−1(x2)+tan⁡−1(y2)+tan⁡−1(z2)=π2\tan^{- 1}\left( \frac{x}{2} \right)+\tan^{- 1}\left( \frac{y}{2} \right)+\tan^{- 1}\left( \frac{z}{2} \right)=\frac{\pi}{2} then xy+yz+zx=xy+yz+zx=

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