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NDA 2020 Maths · Mathematics · Properties of Triangles · medium

For a triangle ABC, consider the following statements: 1. tan A+tan B+tan C=tan Atan Btan C holds for every triangle. 2. cos A+cos B+cos C≤3/2.

Single correct option

For a triangle ABCABC, consider the following statements:
1. tan⁡A+tan⁡B+tan⁡C=tan⁡Atan⁡Btan⁡C\tan A+\tan B+\tan C=\tan A\tan B\tan C holds for every triangle.
2. cos⁡A+cos⁡B+cos⁡C≤32\cos A+\cos B+\cos C\le\dfrac32.
Which of the statements is/are correct?
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
Show answer and solution

Answer: (B) 2 only

The tangent identity needs all three tangents to exist, so it fails for a right-angled triangle where tan⁡90∘\tan 90^\circ is undefined; statement 1 is false as stated. The maximum of cos⁡A+cos⁡B+cos⁡C\cos A+\cos B+\cos C occurs for the equilateral triangle and is 3/23/2, so statement 2 is true.
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Original question written in the NDA 2020 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake