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NDA 2019 Maths · Mathematics · Properties of Triangles · medium

In a triangle ABC which is not right-angled, tan A+tan B+tan C=tan Atan Btan C. 2. In any triangle ABC, sin (A+B)=-sin C.

Single correct option

Consider the following statements:
1. In a triangle ABCABC which is not right-angled, tan⁡A+tan⁡B+tan⁡C=tan⁡Atan⁡Btan⁡C\tan A+\tan B+\tan C=\tan A\tan B\tan C.
2. In any triangle ABCABC, sin⁡(A+B)=−sin⁡C\sin(A+B)=-\sin C.
Which of the statements given above 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: (A) 1 only

Since A+B+C=πA+B+C=\pi, we have tan⁡(A+B)=−tan⁡C\tan(A+B)=-\tan C, which expands to tan⁡A+tan⁡B1−tan⁡Atan⁡B=−tan⁡C\dfrac{\tan A+\tan B}{1-\tan A\tan B}=-\tan C and rearranges to the identity of statement 1, which is true. For statement 2, sin⁡(A+B)=sin⁡(π−C)=sin⁡C\sin(A+B)=\sin(\pi-C)=\sin C, not −sin⁡C-\sin C; it is false. Only statement 1 is correct.
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Original question written in the NDA 2019 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