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NDA 2023 Maths · Mathematics · Trigonometric Equations · medium

The sum of all solutions of tan ^2x=1 in the interval [0,2π) is

Single correct option

The sum of all solutions of tan⁡2x=1\tan^2x=1 in the interval [0,2π)[0,2\pi) is
  1. A. 4π4\pi
  2. B. 2π2\pi
  3. C. 3π3\pi
  4. D. π\pi
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Answer: (A) 4π4\pi

tan⁡x=±1\tan x=\pm1 gives x=π4,3π4,5π4,7π4x=\tfrac\pi4,\tfrac{3\pi}4,\tfrac{5\pi}4,\tfrac{7\pi}4. Their sum is 16π4=4π\tfrac{16\pi}4=4\pi. Taking only tan⁡x=1\tan x=1 (two roots, sum 3π2\tfrac{3\pi}2) or missing the second-quadrant and fourth-quadrant roots leads to the smaller sums.
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