Question

Find the minimum value of ( 4tan²α+9cot²α)

Answer 1

Answer:

12

Step-by-step explanation:

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