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NDA 2025
Exam
NDA
Year
2025
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2025 · Mathematics · Trigonometric Ratios & Identities

Consider the following for the three (03) items that follow: Let p...

Consider the following for the three (03) items that follow:
Let p=tan⁡2α−tan⁡αp = \tan 2\alpha - \tan\alpha and q=cot⁡α−cot⁡2αq = \cot\alpha - \cot 2\alpha.

What is (p/q)(p/q) equal to?

Options

  1. A−tan⁡α⋅tan⁡2α-\tan\alpha \cdot \tan 2\alpha
  2. B−cot⁡α⋅cot⁡2α-\cot\alpha \cdot \cot 2\alpha
  3. Ctan⁡α⋅tan⁡2α\tan\alpha \cdot \tan 2\alpha
  4. Dcot⁡α⋅cot⁡2α\cot\alpha \cdot \cot 2\alpha

Correct answer

C. tan⁡α⋅tan⁡2α\tan\alpha \cdot \tan 2\alpha

Step-by-step solution

Given p=tan⁡2α−tan⁡αp = \tan 2\alpha - \tan\alpha and q=cot⁡α−cot⁡2αq = \cot\alpha - \cot 2\alpha

Expressing qq in terms of tan⁡\tan gives:

q=1tan⁡α−1tan⁡2αq = \dfrac{1}{\tan\alpha} - \dfrac{1}{\tan 2\alpha}

q=tan⁡2α−tan⁡αtan⁡αtan⁡2αq = \dfrac{\tan 2\alpha - \tan\alpha}{\tan\alpha \tan 2\alpha}

Substituting p=tan⁡2α−tan⁡αp = \tan 2\alpha - \tan\alpha into the expression for qq:

q=ptan⁡αtan⁡2αq = \dfrac{p}{\tan\alpha \tan 2\alpha}

Rearranging the terms yields:

pq=tan⁡αtan⁡2α\dfrac{p}{q} = \tan\alpha \tan 2\alpha

Answer: tan⁡α⋅tan⁡2α\tan\alpha \cdot \tan 2\alpha
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