Correct Answer - Option 2 : 2 only

**Calculations :**

Statement 1

2 sin2 θ - cos θ + 4 = 0

⇒ 2 (1 - cos^{2} θ) - cos θ + 4 = 0

⇒ 2 - 2 cos^{2} θ - cos θ + 4 = 0

⇒ 2 cos^{2} θ + cos θ - 6 = 0

⇒ 2 cos^{2} θ + 4 cos θ - 3 cos θ - 6 = 0

⇒ 2 cos θ (cos θ + 2) - 3 (cos θ + 2) = 0

⇒ (2 cos θ - 3)(cos θ + 2) = 0

⇒ cos θ = 3/2 or -2

This statement is incorrect as cos θ lies between -1 and 1

Now statement 2

tan θ + cot θ is more than or equal to 2

(tan θ + cot θ)/2 ≥ √(tan θ .cot θ) (Arithmetic mean ≥ Geometric mean)

(tan θ + cot θ)/2 ≥ 1

Tan θ + cot θ ≥ 2

We can conclude that only statement 2 is correct.

**∴ Option 2 is the correct choice.**