1. **State the problem:** Prove or simplify the expression $$\cos 2A - \cot 2A = \tan A$$. 2. **Recall formulas:** - Double angle formula for cosine: $$\cos 2A = \cos^2 A - \sin^2 A$$ or $$\cos 2A = 1 - 2\sin^2 A$$. - Cotangent definition: $$\cot \theta = \frac{\cos \theta}{\sin \theta}$$. - Tangent definition: $$\tan A = \frac{\sin A}{\cos A}$$. 3. **Rewrite the left side:** $$\cos 2A - \cot 2A = \cos 2A - \frac{\cos 2A}{\sin 2A} = \frac{\cos 2A \sin 2A - \cos 2A}{\sin 2A} = \frac{\cos 2A (\sin 2A - 1)}{\sin 2A}$$. 4. **Use double angle formulas:** - $$\sin 2A = 2 \sin A \cos A$$. - $$\cos 2A = \cos^2 A - \sin^2 A$$. 5. **Substitute:** $$\frac{(\cos^2 A - \sin^2 A)(2 \sin A \cos A - 1)}{2 \sin A \cos A}$$. 6. **Check if this equals $$\tan A = \frac{\sin A}{\cos A}$$:** Multiply both sides by denominator to compare: $$ (\cos^2 A - \sin^2 A)(2 \sin A \cos A - 1) \stackrel{?}{=} 2 \sin^2 A \cos A $$ 7. **Test with specific values or simplify further:** This expression is not generally equal to $$\tan A$$ for all $$A$$. **Conclusion:** The equation $$\cos 2A - \cot 2A = \tan A$$ is not an identity and does not hold true for all $$A$$. **Final answer:** The given equation is false in general.