What is the minimum value of 9 tan^2 θ + 4 cot^2 θ

• Quant Trigo

What is the minimum value of 9 tan² θ + 4 cot² θ

1. 6
2. 9
3. 12
4. 13

Answer

The minimum value of a tan² θ + b cot² θ is 2√(ab)

Here, a = 9 and b = 4

So, 2√(ab) = 2√(9×4) = 12

Arithmetic mean is always greater than or equal to geometric mean.

AM ≥ GM or (a+b)/2 ≥ √(ab)

Minimum value of a+b = 2√(ab)

Arithmetic mean = 9 tan² θ + 4 cot² θ

Geometric mean = √(9 tan²θ × 4 cot²θ)

tan²θ = 1/cot²θ

So, Geometric mean = √(9 × 4) = 6

Minimum value = 2 × GM = 12

The correct option is C.