Correct Answer - Option 2 : 2 only
Calculations :
Statement 1
2 sin2 θ - cos θ + 4 = 0
⇒ 2 (1 - cos2 θ) - cos θ + 4 = 0
⇒ 2 - 2 cos2 θ - cos θ + 4 = 0
⇒ 2 cos2 θ + cos θ - 6 = 0
⇒ 2 cos2 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.