Question

If 4cot²θ + (sin²θ –1)(1 + cot²θ) - 2 = 7, 0° ≤ θ 90° then θ is:
1. 15° 
2. 30° 
3. 45° 
4. 60° 

Answer 1

Correct Answer - Option 2 : 30° 

Given :

(sin2θ –1)(1 + cot2θ) + 4cot2θ - 2 = 7

Formula used :

cos²θ + sin²θ = 1

Calculations :

(sin2θ –1)(1 + cot2θ) + 4cot2θ - 2 = 7

⇒ sin2θ + sin2θ.cot2θ - 1 - cot2θ + 4cot2θ - 2 = 7

⇒ sin2θ + cos2θ - 1 - cot2θ + 4cot2θ - 2 = 7   (∵ cot2θ = cos2θ/sin²θ) 

⇒ 1 - 1 - cot2θ + 4cot2θ = 9 

⇒ 3cot2θ = 9 

⇒ cot2θ = 3 

⇒ cotθ = √3 

⇒ θ = 30°

∴ Option 2 will be the correct choice.