What is the value of (1 – cos2θ) cosec2θ?

1.	What is the value of (1 – cos2θ) cosec2θ?
Answer» To find: (1 – cos²θ) cosec²θ ∵ cosecθ = \(\frac{1}{sinθ}\) ∴ cosec²θ = \(\frac{1}{sin^2θ}\) ⇒ (1 – cos²θ) cosec²θ = (1 – cos²θ)\(\frac{1}{sin^2θ}\) ....(1) ∵ sin² θ + cos² θ = 1 ∴ sin² θ = 1 – cos² θ ⇒ from (i), we have ⇒ (1 – cos²θ) cosec²θ = sin² θ \(\frac{1}{sin^2θ}\) = 1

Answer»

To find: (1 – cos²θ) cosec²θ

∵ cosecθ = \(\frac{1}{sinθ}\)

∴ cosec²θ = \(\frac{1}{sin^2θ}\)

⇒ (1 – cos²θ) cosec²θ = (1 – cos²θ)\(\frac{1}{sin^2θ}\) ....(1)

∵ sin² θ + cos² θ = 1

∴ sin² θ = 1 – cos² θ

⇒ from (i), we have

⇒ (1 – cos²θ) cosec²θ = sin² θ \(\frac{1}{sin^2θ}\) = 1

Discussion