Question

Question࿐

➢If a cotθ + b cosecθ = p and b cotθ + a cosecθ = q , than p² - q² is:-​

Answer 1

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

Answer 2

Answer:

Given, acotθ+bcscθ=p

            bcotθ+acscθ=q

⇒p2−q2=(p−q)(p+q)

                    =[acotθ+bcscθ−bcotθ−acscθ][acotθ+bcscθ+bcotθ+acscθ]

                    =[cotθ(a−b)−cscθ(a−b)][cotθ(a+b)+cscθ(a+b)]

                    =(a−b)(cotθ−cscθ)(a+b)(cotθ+cscθ)

                    =(a2−b2)(cot2θ−csc2θ)

                    =(−1)(a2−b2)[∵csc2θ−cot2θ=1]

                    =(b2−a2)

❣✌