If \(\rm \cos27°=\frac{P}{q}\), then find the value of cosec 27° - cos 63°.

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Question

This question was previously asked in

SSC CGL 2024 Tier-II Official Paper-I (Held On: 20 Jan, 2025)

Answer 1

Given:

cos 27° = p/q

Formula used:

cosec A = 1 / sin A

cos (90° - A) = sin A

Calculations:

⇒ cosec 27° = 1 / sin 27°

⇒ cos 63° = sin 27° (since cos (90° - A) = sin A)

⇒ cosec 27° - cos 63° = 1 / sin 27° - sin 27°

⇒ cosec 27° - cos 63° = (1 - sin² 27°) / sin 27°

⇒ cosec 27° - cos 63° = cos² 27° / sin 27°

Now, using the given cos 27° = p/q, we get:

⇒ cos² 27° = (p² / q²)

⇒ sin² 27° = 1 - cos² 27° = 1 - (p² / q²)

⇒ sin 27° = √(q² - p²) / q

Substituting this into the equation for cosec 27° - cos 63°:

⇒ cosec 27° - cos 63° = (p² / q²) / (√(q² - p²) / q)

⇒ cosec 27° - cos 63° = p² / (q√(q² - p²))

∴ The correct answer is option (2).

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