Mathematics
If a cot θ + b cosec θ = p and b cot θ + a cosec θ = q, then p2 − q2 is equal to :
a2 − b2
b2 − a2
a2 + b2
b − a
Answer
⇒ a cot θ + b cosec θ = p
p2 = (a cot θ + b cosec θ)2
p2 = (a2 cot2 θ + b2 cosec2 θ + 2ab cot θ cosec θ)
⇒ b cot θ + a cosec θ = q
q2 = (b cot θ + a cosec θ)2
q2 = (b2 cot2 θ + a2 cosec2 θ + 2ab cot θ cosec θ)
⇒ p2 − q2 = (a2 cot2 θ + b2 cosec2 θ + 2ab cot θ cosec θ) - (b2 cot2 θ + a2 cosec2 θ + 2ab cot θ cosec θ)
= (a2 cot2 θ + b2 cosec2 θ + 2ab cot θ cosec θ - b2 cot2 θ - a2 cosec2 θ - 2ab cot θ cosec θ)
= (a2 cot2 θ + b2 cosec2 θ - b2 cot2 θ - a2 cosec2 θ )
= a2 (cot2 θ - cosec2) + b2 (cosec2 θ - cot2 θ )
= a2 (-1) + b2 (1)
= b2 - a2.
Hence, option 2 is the correct option.