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NDA 2 2025 Maths, question 26

Topic: Trigonometry

Passage · Q25 to Q26

Let sin⁡A+sin⁡B=p\sin A+\sin B=p and cos⁡A+cos⁡B=q\cos A+\cos B=q.

Q26

What is p2−q2p2+q2\frac{p^{2}-q^{2}}{p^{2}+q^{2}} equal to?

  1. (a) cos⁡(A+B)\cos (A+B)
  2. (b) cos⁡(A−B)\cos (A-B)
  3. (c) cos⁡(π2−A−B)\cos \left( \frac{\pi}{2}-A-B \right)
  4. (d) cos⁡(π−A−B)\cos (\pi-A-B)Correct answer

Solution.

Answer: (d) cos⁡(π−A−B)\cos (\pi-A-B)

  1. Using the same identities, the common factor cancels and the ratio becomes sin²((A+B)/2)−cos²((A+B)/2)=−cos(A+B)=cos(π−A−B).
  2. This holds wherever the stated ratio is defined.