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

1.	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» 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

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»

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