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NDA 2023 Maths · Mathematics · Trigonometric Functions and Identities · hard

For all admissible θ, (sin 3θ)/(sin θ)-(cos 3θ)/(cos θ) equals

Single correct option

For all admissible θ\theta, sin⁡3θsin⁡θ−cos⁡3θcos⁡θ\dfrac{\sin3\theta}{\sin\theta}-\dfrac{\cos3\theta}{\cos\theta} equals
  1. A. 00
  2. B. 22
  3. C. 11
  4. D. 2sin⁡2θ2\sin2\theta
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Answer: (B) 22

Combine the fractions: numerator sin⁡3θcos⁡θ−cos⁡3θsin⁡θ=sin⁡(3θ−θ)=sin⁡2θ\sin3\theta\cos\theta-\cos3\theta\sin\theta=\sin(3\theta-\theta)=\sin2\theta; denominator sin⁡θcos⁡θ=12sin⁡2θ\sin\theta\cos\theta=\tfrac12\sin2\theta. The ratio is 22, a constant. Writing a plus sign between the fractions would instead give sin⁡4θ12sin⁡2θ=4cos⁡2θ\dfrac{\sin4\theta}{\tfrac12\sin2\theta}=4\cos2\theta.
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Original question written in the NDA 2023 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