JEE Main 1987 Mathematics question on Trigonometry

tan x = x requires the line y = x to intersect y = tan x. In (0, π/2), tan x > 0 but tan x → +∞ as x → π/2⁻, yet tan x starts at 0 and the line y = x also starts at 0 — they coincide only at x = 0 (trivial), and for x > 0 in this interval tan x > x but there's no crossing away from 0. In (π/2, π), tan x is negative while x is positive, so no solution. In (π, 3π/2), tan x goes from 0 (at x = π, tan π = 0 < π) and increases to +∞ as x → 3π/2⁻, so by the intermediate value theorem tan x − x crosses zero somewhere in (π, 3π/2), giving the smallest positive root.