If cos 24° = m/n, then find the value of (cosec 24° - cos 66°).

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Given: cos 24° = m/n We need to find the value of (cosec 24° - cos 66°). Step 1: Express cosec 24° in terms of sin 24°. cosec 24° = 1/sin 24° Step 2: Note that cos 66° = sin 24° (since 66° + 24° = 90°, sin θ = cos (90° - θ)) Therefore, cos 66° = sin 24° Step 3: Substitute cos 66° = sin 24° in the expression: cosec 24° - cos 66° = (1/sin 24°) - sin 24° Step 4: Let sin 24° = s. Expression becomes: (1/s) - s = (1 - s²)/s Step 5: Use the Pythagorean identity: sin² 24° + cos² 24° = 1 => s² + (m/n)² = 1 => s² = 1 - (m²/n²) = (n² - m²)/n² => s = sqrt((n² - m²)/n²) = sqrt(n² - m²)/n Step 6: Substitute s and s² into the expression: (1 - s²)/s = (1 - (n² - m²)/n²) / (sqrt(n² - m²)/n) = ((n²/n²) - (n² - m²)/n²) / (sqrt(n² - m²)/n) = (m²/n²) / (sqrt(n² - m²)/n) = (m²/n²) * (n / sqrt(n² - m²)) = (m² / n) * (1 / sqrt(n² - m²)) Step 7: The expression simplifies to: (m²) / (n * sqrt(n² - m²)) Step 8: Check options: Option 3 matches this expression. However, the problem states cos 24° = m/n, so m < n (since cos 24° < 1), so n² - m² > 0, so sqrt(n² - m²) is valid. Therefore, the correct answer is option 3: (m²) / (n√(n² - m²)) Rechecking options, option 3 is (m²) / (n√(n² - m²)) which matches our derived expression. Hence, option 3 is correct.

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