If cosecθ = 2x and cotθ = 2/x ,find the value of 2(x^2-1/x^2)

Question

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

Answer 1

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 = cosec² θ – cot² θ

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