1. **State the problem:** Prove the identity \( \tan A - \cot A = 2 \tan(2A) \). 2. **Rewrite cotangent:** Recall that \( \cot A = \frac{1}{\tan A} \). 3. **Express left side with common denominator:** $$ \tan A - \cot A = \tan A - \frac{1}{\tan A} = \frac{\tan^2 A - 1}{\tan A} $$ 4. **Use the double-angle formulas:** Recall that $$ \tan(2A) = \frac{2 \tan A}{1 - \tan^2 A} $$ 5. **Manipulate numerator \( \tan^2 A - 1 \) to match denominator \(1-\tan^2 A\):** $$ \tan^2 A - 1 = - (1 - \tan^2 A) $$ 6. **Rewrite the left side using this:** $$ \frac{\tan^2 A - 1}{\tan A} = \frac{- (1 - \tan^2 A)}{\tan A} = - \frac{1 - \tan^2 A}{\tan A} $$ 7. **Compare with \( 2 \tan(2A) \):** We have $$ 2 \tan(2A) = 2 \cdot \frac{2 \tan A}{1 - \tan^2 A} = \frac{4 \tan A}{1 - \tan^2 A} $$ 8. **Check if \( \tan A - \cot A = 2 \tan(2A) \):** From step 6, left side is $$ - \frac{1 - \tan^2 A}{\tan A} $$ which is **not equal** to right side \( \frac{4 \tan A}{1 - \tan^2 A} \). 9. **Conclusion:** The given identity \( \tan A - \cot A = 2 \tan(2A) \) is incorrect as stated. Instead, check the correct identity $$ \tan A - \cot A = -2 \cot(2A) $$ or $$ \tan A - \cot A = \frac{2 \tan(2A)}{1 - \tan^2(2A)} $$ depending on the context. Hence, the original formula cannot be proven true as is.