Question

If a cot θ + b cosec θ = p and b cot θ + a cosec θ = q, then p²− q² = 

A. a²− b² 

B. b²− a² 

C. a² + b² 

D. b − a

Answer 1

Given: a cot θ + b cosec θ = p 

Squaring both sides, we get 

(a cot θ + b cosec θ)² = p²

⇒ a² cot²θ + b² cosec²θ + 2ab cotθ cosecθ = p² ……(i) 

and b cotθ + a cosecθ = q 

Squaring both sides, we get 

(b cot θ + a cosec θ)² = q² 

⇒ b² cot²θ + a² cosec²θ + 2ab cotθ cosecθ = q² ……(ii) 

To find: p² – q² 

Subtracting (ii) from (i), we get 

a² cot²θ + b² cosec²θ + 2ab cotθ cosecθ – b² cot²θ – a² cosec²θ – 2ab cotθ cosecθ = p² – q² 

⇒ P² – q² = a² (cot²θ – cosec²θ) + b² (cosec²θ – cot²θ) 

= a² ( – 1) + b² (1) [∵1 = cosec²θ – cot²θ] 

= b² – a²