Prove and apply the Pythagorean identities. sin2(𝐴) + cos2(𝐴) = 1 tan2(𝐴) + 1 = sec2(𝐴) cot2(𝐴) + 1 = cosec2(𝐴)

Prove that \(\cot^2(\theta) + 1 = \operatorname{cosec}^2(\theta)\) for \(\theta \in \mathbb{R}\), \(\theta \neq n\pi\) where \(n \in \mathbb{Z}\).

Prove \(\tan^2(\theta) + 1 = \sec^2(\theta)\) using the identity \(\sin^2(\theta) + \cos^2(\theta) = 1\).

A student is investigating the relationship between trigonometric functions. They are given a right-angled triangle with hypotenuse r, opposite side y, and adjacent side x, where x > 0, y > 0, and r > 0. (a) Using the diagram, prove that cot²(θ) + 1 = cosec²(θ). (3 marks) (b) Hence, find the exact value of cot(θ) when cosec(θ) = 13/5 and θ is acute. (2 marks)

The diagram shows a right-angled triangle with angle θ at the origin, where the hypotenuse has length 1. Given that cos(θ) = a and sin(θ) = b, which expression correctly represents tan²(θ) + 1?