If <span class="math-tex">\(\rm \frac{\tan x}{2} = \frac{\tan y}{3} = \frac{\tan z}{5}\)</span> and x + y + z = π, then the value of tan<sup>2</sup> x + tan<sup>2</sup> y + tan<sup>2</sup> z is:
Let \( \tan x = 2k, \; \tan y = 3k, \; \tan z = 5k \). We know that when \( x + y + z = \pi \), \[ \tan x + \tan y + \tan z = \tan x \tan y \tan z \] \[ \Rightarrow 2k + 3k + 5k = (2k)(3k)(5k) \] \[ \Rightarrow 10k = 30k^3 \] or \[ k^2 = \frac{1}{3} \] Hence: \[ \tan^2 x + \tan^2 y + \tan^2 z = 4k^2 + 9k^2 + 25k^2 \] \[ = 38k^2 = \frac{38}{3} \]
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