Question

if tanθ + cotθ = 4, then the ratio of 3(tan²θ + cot²θ) to (2cosec²θ sec²θ - 4) will be:
1. 5 : 4
2. 4 : 3
3. 3 : 2
4. 3 : 4

Answer 1

Correct Answer - Option 3 : 3 : 2

Given:

tanθ + cotθ = 4

Formula:

tan² θ + 1 = sec² θ

cot² + 1 = cosec² θ

Calculation:

⇒ tanθ + cotθ = 4

⇒ tan θ + cot θ)² = tan²θ + 2tanθ.cotθ + cot²θ = 16

⇒ tan2θ + 2tanθ.cotθ + cot2θ = 16

⇒ tan²θ + cot²θ = 16 - 2 = 14

Then,

⇒ ? = (2cosec2θ sec2θ - 4)

⇒ ? = 2(1 + cot²θ)(1 + tan²θ) - 4

⇒ ? = 2(1 + tan²θ + cot²θ + cot²θ.tan²θ) - 4

⇒ ? = 2(1 + 14 + 1) - 4

⇒ ? = 28

Then,

⇒ 3(tan2θ + cot2θ) : (2cosec2θ sec2θ - 4) = 3 × 14 : 28 = 3 : 2

∴ 3(tan2θ + cot2θ) : (2cosec2θ sec2θ - 4) = 3 : 2