If cos 24° = m⁄n, then the value of (cosec 24° − cos 66°) is:

Correct — C. The whole item turns on one swap: cos 66° = sin 24°, because 66° and 24° add to 90°.

cosec 24° − cos 66° = 1⁄sin 24° − sin 24°
= (1 − sin²24°)⁄sin 24°
= cos²24° ⁄ sin 24°

cos 24° = m⁄n, so sin 24° = √(n² − m²)⁄n and cos²24° = m²⁄n².

Substituting: (m²⁄n²) × (n ⁄ √(n² − m²))
= m² ⁄ (n√(n² − m²)) → option (c)

Why the others are wrong

• (a)
  The radical is right but the letters outside it are the wrong way round. cos 24° = m⁄n squares to m²⁄n², so m² belongs on top and n in the denominator, not n² over m.
• (b)
  The radicand is reversed. sin²24° = 1 − cos²24° = 1 − m²⁄n², which gives √(n² − m²)⁄n. Writing √(m² − n²) is negative under the root, since cos 24° ≈ 0.914 means m < n.
• (d)
  This swaps m and n everywhere — the expression you would write if the question had said cos 24° = n⁄m. The paper puts m on top, and m⁄n is a cosine of an acute angle, so m < n.

Two identities do all the work. The first is complementary: cos(90° − θ) = sin θ, so a cosine of 66° is a sine of 24° and the expression collapses to one angle.

The second is Pythagorean: sin²θ + cos²θ = 1. Once everything is at 24°, cosec 24° − sin 24° becomes (1 − sin²24°)⁄sin 24° = cos²24°⁄sin 24°.

The last step is bookkeeping. A given cos θ = m⁄n fixes the adjacent side as m and the hypotenuse as n, so the opposite side is √(n² − m²) and every other ratio follows from it.

Key facts

• cos(90° − θ) = sin θ, so cos 66° = sin 24° and sin 66° = cos 24°.
• cosec θ − sin θ = cos²θ ⁄ sin θ, and sec θ − cos θ = sin²θ ⁄ cos θ.
• If cos θ = m⁄n for an acute θ, then sin θ = √(n² − m²)⁄n, taking the positive root.

Common traps

• Subtracting cos 66° as a cosine instead of converting it to sin 24° first
• Taking sin 24° = √(m² − n²)/n, which puts a negative quantity under the root
• Squaring m/n to n²/m² by reflex when the fraction gets inverted later in the working

No directly related past PYQ was found.