If cosecθ = 2x and cotθ = \(\frac{2}{x}\) , find the value of 2 \

1.	If cosecθ = 2x and cotθ = \(\frac{2}{x}\) , find the value of 2 \(\Big(x^2-\frac{1}{x^2}\Big)\)
Answer» Given: cosecθ = 2x ⇒ x = \(\frac{ cosec θ}{2}\) ⇒ x² = \(\frac{ cosec^2θ}{4}\) .........(1) And cotθ = \(\frac{2}{x}\) ⇒ x = \(\frac{2}{cotθ}\) ⇒ x²= \(\frac{4}{cot^2θ}\) ⇒ \(\frac{1}{x^2}\) = \(\frac{cot^2θ}{4}\) .....(ii) To find: \(2\Big(x^2-\frac{1}{x^2}\Big)\) Consider \(2\Big(x^2-\frac{1}{x^2}\Big)\) = \(2\Big(\frac{cosec^2 θ}{4}-\frac{1}{x^2}\Big)\) [Using (i)] = \(2\Big(\frac{cosec^2 θ}{4}-\frac{cot^2 θ}{4}\Big)\) [Using (ii)] = \(2\Big(\frac{cosec^2 θ-cot^2 θ}{4}\Big)\) = \(\frac{1}{2}\) (cosec²θ – cot²θ) Now, as 1 + cot²θ = cosec²θ ⇒ 1 = cosec²θ – cot²θ ⇒ \(2\Big(x^2-\frac{1}{x^2}\Big)\) = \(\frac{1}{2}\) (cosec²θ – cot²θ) = \(\frac{1}{2}\)

If cosecθ = 2x and cotθ = \(\frac{2}{x}\) , find the value of 2 \(\Big(x^2-\frac{1}{x^2}\Big)\)

Answer»

Given: cosecθ = 2x

⇒ x = \(\frac{ cosec θ}{2}\)

⇒ x² = \(\frac{ cosec^2θ}{4}\) .........(1)

And cotθ = \(\frac{2}{x}\)

⇒ x = \(\frac{2}{cotθ}\)

⇒ x²= \(\frac{4}{cot^2θ}\)

⇒ \(\frac{1}{x^2}\) = \(\frac{cot^2θ}{4}\) .....(ii)

To find: \(2\Big(x^2-\frac{1}{x^2}\Big)\)

Consider \(2\Big(x^2-\frac{1}{x^2}\Big)\) = \(2\Big(\frac{cosec^2 θ}{4}-\frac{1}{x^2}\Big)\) [Using (i)]

= \(2\Big(\frac{cosec^2 θ}{4}-\frac{cot^2 θ}{4}\Big)\) [Using (ii)]

= \(2\Big(\frac{cosec^2 θ-cot^2 θ}{4}\Big)\) = \(\frac{1}{2}\) (cosec²θ – cot²θ)

Now, as 1 + cot²θ = cosec²θ

⇒ 1 = cosec²θ – cot²θ

⇒ \(2\Big(x^2-\frac{1}{x^2}\Big)\) = \(\frac{1}{2}\) (cosec²θ – cot²θ) = \(\frac{1}{2}\)