Prove and apply multi-angle trigonometric identities up to angles of 4𝑥 using the identities listed above, e.g. cos(4𝑥) = 8 cos4(𝑥) − 8 cos2(𝑥) + 1 and cosec(2𝑥) − cot(2𝑥) = tan(𝑥). Specialist Mathematics 2025 v1.4

Prove that \[\csc(2x) - \cot(2x) = \tan(x)\] using the identities \[\sin(2x) = 2\sin(x)\cos(x), \quad \cos(2x) = \cos^2(x) - \sin^2(x), \quad \text{and} \quad \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}.\]

Prove that \(\csc(2x) - \cot(2x) = \tan(x)\) for \(x \neq n\pi\), where \(n \in \mathbb{Z}\).

Prove that \(\sec(2x) + \tan(2x) = \frac{\cos(x) + \sin(x)}{\cos(x) - \sin(x)}\).

Which of the following is the correct expression for \(\tan(3x)\) in terms of \(\tan(x)\)?