Question

The value of \(\frac{tan^2 30^\circ+sin^290^\circ+cot^260^\circ+sin^230^\circ cos^245^\circ}{sin60^\circ cos30^\circ-cos60^\circ sin30^\circ}\)
1. \(\frac{47}{12}\)
2. \(\frac{37}{12}\)
3. \(\frac{25}{12}\)
4. \(\frac{43}{12}\)

Answer 1

Correct Answer - Option 4 : \(\frac{43}{12}\)

Given:

? = \(\frac{tan^2 30^\circ+sin^290^\circ+cot^260^\circ+sin^230^\circ cos^245^\circ}{sin60^\circ cos30^\circ-cos60^\circ sin30^\circ}\)

Concept:

tan30 = 1/√3 and sin 90 = 1

cot 60 = 1/√3 and sin 30 = 1/2

cos 45 = 1/√2 and sin 60 = √3/2

cos 30 = √3/2 and cos 60 = 1/2

Calculation:

⇒ ? = [(1/√3)² + 1² + (1/√3)² + (1/2)² × (1/√2)²]/[(√3/2) × (√3/2) - 1/2 × 1/2]

⇒ ? = [1/3 + 1 + 1/3 + 1/4 × 1/2]/[3/4 - 1/4]

⇒ ? = [43/24] × 2

∴ ? = 43/12