1. The problem is to prove the identity: $$\cos 2A - \cot 2A = \tan 2A$$. 2. Recall the definitions and identities: - $$\tan x = \frac{\sin x}{\cos x}$$ - $$\cot x = \frac{\cos x}{\sin x}$$ - $$\cos 2A$$ is a standard trigonometric function. 3. Rewrite the left side using the definition of cotangent: $$\cos 2A - \cot 2A = \cos 2A - \frac{\cos 2A}{\sin 2A}$$ 4. Find a common denominator $$\sin 2A$$: $$= \frac{\cos 2A \sin 2A}{\sin 2A} - \frac{\cos 2A}{\sin 2A} = \frac{\cos 2A \sin 2A - \cos 2A}{\sin 2A}$$ 5. Factor out $$\cos 2A$$ in the numerator: $$= \frac{\cos 2A (\sin 2A - 1)}{\sin 2A}$$ 6. The right side is $$\tan 2A = \frac{\sin 2A}{\cos 2A}$$. 7. The left side expression $$\frac{\cos 2A (\sin 2A - 1)}{\sin 2A}$$ does not simplify to $$\frac{\sin 2A}{\cos 2A}$$ in general. 8. Therefore, the identity $$\cos 2A - \cot 2A = \tan 2A$$ is not true for all $$A$$. Final answer: The given identity is false in general.