Question

If tanα + cosβ = k and secβ + cotα = l if α ≠ β, then find the value of 1 - cot²α(k/l)²

1. sin²α 
2. sin2β
3. cos²α 
4. cot²β 

Answer 1

Correct Answer - Option 2 : sin2β

Given:

tanα + cosβ = k

secβ + cotα = l

Concept used:

(1/cotθ) = tanθ

(1/cosecθ) = sinθ

1 + tan²θ = sec²θ

Calculations:

secβ + cotα = l

⇒ (1/cosβ) + (1/tanα) = l

⇒ (tanα + cosβ)/(tanα × cosβ) = l

⇒ k/(tanα × cosβ) = l

⇒ k/l = tanα × cosβ      ....(1)

Square the equation no. (1)

⇒ (k/l)² = tan²α × cos²β      ....(2)

Multiply both side of equation no. (2) with cot²α 

⇒ cot²α  × (k/l)² = tan²α × cos²β × cot²α 

⇒ cot²α × (k/l)² = cos²β   

Now,  

1 - cot2α(k/l)² = 1 - cos2β

⇒ sin2β  

∴ The correct choice will be option 2.