Which of the following is/are true?

i. tan2 θ  + cot2 θ ≤ 2 

ii. tan2 θ  + cot2 θ ≥ 2 

iii. tan2 θ  + cot2 θ ≤ 1 

Concept:

AM-GM-HM Inequality: AM ≥ GM ≥ HM.

Let two positive numbers are a and b:

⇒ \(\rm \frac{a+b}{2}\geq \sqrt[2]{ab}\geq \frac{2ab}{a+b}\)

Calculation:

As we know, AM ≥ GM

⇒ \(\rm \frac{\tan^2 \theta+\cot^2 \theta }{2}\geq \sqrt[2]{\tan^2 \theta \times\cot^2 \theta}\)

⇒ \(\rm \frac{\tan^2 \theta+\cot^2 \theta }{2}\geq 1\)

∴ tan2 θ  + cot2 θ ≥ 2

Let, θ = 45° 

\(\rm tan^2(45)+cot^2(45)=1+1=2\)        ....(1)

Let, θ = 60° 

\(\rm tan^2(60)+cot^2(60)=(\sqrt3)^2+(\frac{1}{\sqrt3})^2\)

= 3 + \(\frac 1 3\) > 2        ....(2)

From (1) and (2) we can say that, 

tan2 θ  + cot2 θ ≥ 2

Hence, option (2) is correct.