Question

If the value of sec²2θ – 1 = 3, then find the value of 3 sin θ + {2 tan θ/(1 + tan² θ)}² + cosec² θ – cot² θ – 4 sin³ θ?
1. 5
2. 11/4
3. 3
4. 11/2

Answer 1

Correct Answer - Option 2 : 11/4

Given:

sec22θ – 1 = 3

Concept used:

sin 3A = 3 sin A – 4 sin3 A

sin 2A = {2 tan A/(1 + tan2 A)}2 

1 + cot2A = cosec2A  

cosec2A  – cot2A = 1

cos 60° = 1/2

Calculation:

sec22θ – 1 = 3

⇒ sec22θ = 4

Taking square root on both sides,

⇒ sec 2θ = 2

⇒ cos 2θ = 1/2

We knoe that, cos 60° = 1/2

⇒ 2θ = 60° 

⇒ θ = 30° 

(3 sin θ – 4 sin3 θ) + {2 tan θ/(1 + tan2 θ)}2 + cosec2 θ  - cot2 θ = sin 3θ + sin² 2θ + 1

⇒ sin (3 × 30°) + sin² (2 × 30°) + 1

⇒ sin 90° + sin² 60° + 1

⇒ 1 + (3/4) + 1

⇒ 2 + (3/4)

⇒ (8 + 3)/4

⇒ 11/4

∴ The value is 11/4.