Q190 / 269 · NDA 2025 (Phase 1) · Mathematics
Consider the following for the three (03) items that follow:
Let $p = \tan 2\alpha - \tan\alpha$ and $q = \cot\alpha - \cot 2\alpha$.
What is $(p/q)$ equal to?
Let $p = \tan 2\alpha - \tan\alpha$ and $q = \cot\alpha - \cot 2\alpha$.
What is $(p/q)$ equal to?
- A$-\tan\alpha \cdot \tan 2\alpha$
- B$-\cot\alpha \cdot \cot 2\alpha$
- Correct answerC$\tan\alpha \cdot \tan 2\alpha$
- D$\cot\alpha \cdot \cot 2\alpha$
Solution
Given $p = \tan 2\alpha - \tan\alpha$ and $q = \cot\alpha - \cot 2\alpha$
Expressing $q$ in terms of $\tan$ gives:
$q = \dfrac{1}{\tan\alpha} - \dfrac{1}{\tan 2\alpha}$
$q = \dfrac{\tan 2\alpha - \tan\alpha}{\tan\alpha \tan 2\alpha}$
Substituting $p = \tan 2\alpha - \tan\alpha$ into the expression for $q$:
$q = \dfrac{p}{\tan\alpha \tan 2\alpha}$
Rearranging the terms yields:
$\dfrac{p}{q} = \tan\alpha \tan 2\alpha$
Answer: $\tan\alpha \cdot \tan 2\alpha$
Expressing $q$ in terms of $\tan$ gives:
$q = \dfrac{1}{\tan\alpha} - \dfrac{1}{\tan 2\alpha}$
$q = \dfrac{\tan 2\alpha - \tan\alpha}{\tan\alpha \tan 2\alpha}$
Substituting $p = \tan 2\alpha - \tan\alpha$ into the expression for $q$:
$q = \dfrac{p}{\tan\alpha \tan 2\alpha}$
Rearranging the terms yields:
$\dfrac{p}{q} = \tan\alpha \tan 2\alpha$
Answer: $\tan\alpha \cdot \tan 2\alpha$
Hence, the correct answer is option: 'C'.