Question

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

Options

• a² − b²

• b² − a²

• a² + b²

•  b − a

Answer 1

Given: 

`a cotθ+b cosecθ=P,`

`b cotθ+a cosecθ=q `

Squaring both the equations and then subtracting the second from the first, we have

`(p)^2-(q)^2=(a cot θ+b.cosecθ)^2-(b cot θ+a cosecθ)^2`

`=(a^2cot^θ+b^2 cosec^2θ+2.a cotθ.b cosecθ)-(b^2 cot^2θ+a^2 cosec^2θ+2 cotθ.a cosecθ)`

`=a^2 cot^2θ+b^2 cosec^2θ+2 ab cotθ cosecθ-b^2 cot^2θ-a^2cosec^2θ-2ab cotθcosecθ`

`⇒a^2 cot^2θ+b^2 cosec^2θ-b^2 cot^2θ-a^2 cosec^2θ`

`⇒(b^2 cosec^θ-b^2 cot^2 θ)+(-a^2 cosec^2θ+a^2 cot^2θ)=p^2-q^2`

`⇒b^2(cosec^2θ-cot^2θ)-a^2(cosec^θ-cot^2θ)=p^2-q^2`

`⇒b^2(1)-a^2(1)=p^2-q^2`

`⇒b^2-a^2=p^2-q^2` 

`⇒p^2-q^2=b^2-a^2`