Question

\(24\left(\sin^2 60^\circ + \cos^2 30^\circ + \cot^2 60^\circ + \csc^2 45^\circ\right) =\)

• A. \(23\)
• B. \(92\)
• C. \(46\)
• D. \(69\)

Hint:

When multiplying a sum of fractions by a constant, it often helps to multiply each term separately first, then add. Choosing the multiplier (here 24) as the LCD of all denominators (4, 3, etc.) is deliberate in exam problems to produce a clean integer answer.

Answer 1

The Correct Option is B

Concept:
Use the standard trigonometric values:

Step 1: Compute each squared value.
align* ^2 60^ &= (32)^2 = 34.
^2 30^ &= (32)^2 = 34.
^2 60^ &= (13)^2 = 13.
^2 45^ &= (1 45^)^2 = (11/2)^2 = (2)^2 = 2. align*

Step 2: Add the four values.

Find a common denominator (LCD of 4, 4, 3, 1 is 12):

Step 3: Multiply by 24.


Step 4: Verify term by term.

• \(24 \times \dfrac{3}{4} = 18\).

• \(24 \times \dfrac{3}{4} = 18\).

• \(24 \times \dfrac{1}{3} = 8\).

• \(24 \times 2 = 48\).

• Total: \(18 + 18 + 8 + 48 = 92\).