If p tan (θ - 30°) = q tan (θ + 120°), then what is (p + q) / (p - q) equal to?

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NDA-II 2024 (Maths) Official Paper (Held On: 01 Sept, 2024)
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  1. sin 2θ
  2. cos 2θ
  3. 2 sin 2θ
  4. 2 cos 2θ

Answer (Detailed Solution Below)

Option 4 : 2 cos 2θ
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Explanation:

Given:

p tan (θ - 30°) = q tan (θ + 120°)

⇒ p\q = \(\frac{tan(\theta + 120°)}{tan(\theta -30°)}\)

\(\frac{sin(\theta+120)cos(\theta -30)}{cos(\theta+120)sin(\theta-30)}\)

⇒ \(\frac{p+q}{p-q} = \)\(\frac{sin(\theta+120)cos(\theta -30)+cos(\theta+120)sin(\theta-30)}{sin(\theta+120)cos(\theta -30)-cos(\theta+120)sin(\theta-30)}\)

\(\frac{sin[(\theta+120)+(\theta-30)]}{sin[(\theta+120)-(\theta-30)]}\)

\(\frac{sin(90+2\theta)}{sin 150}\)

⇒ \(\frac{p+q}{p-q} = \frac{sin(90+2\theta)}{\frac{1}{2}}\)

= 2 cos2θ

∴ Option (d) is correct

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