Verify the identity by transforming the left-hand side into the right-hand side.

(1 + cos² 3θ) / (sin² 3θ) = 2 csc² 3θ - 1

(1 + cos² 3θ) / (sin² 3θ) = 2 csc² 3θ - 1

Starting with the left: Note that cos²θ + sin²θ = 1.

In the same way: cos²3θ + sin²3θ = 1

Therefore cos²3θ = 1 - sin²3θ

From the top: (1 + cos² 3θ) = 1 + 1 - sin²3θ = 2 - sin²3θ

(1 + cos² 3θ) / (sin² 3θ) = (2 - sin²3θ) / (sin² 3θ) = 2 / sin² 3θ - sin²3θ / sin²3θ

= 2 / sin² 3θ - 1; But 1 / sinθ = csc θ, Similarly 1/sin3θ = csc3θ

= 2 * (1/sin 3θ) ² - 1

= 2csc²3θ - 1. Therefore LHS = RHS. QED.