Prove that: `tan^(-1)(1/2tan2A)+tan^(-1)(cota)+tan^(-1)(cot^3A)={0`` when pi/4 Ale pi/2` and `pi`,`` when ``0 le A le pi/4`}`

Explanation

To prove that

\[ \tan^{-1}\left(\frac{1}{2}\tan 2A\right) + \tan^{-1}(\cot A) + \tan^{-1}(\cot^3 A) = 0 \]

for \( \frac{\pi}{4} < A < \frac{\pi}{2} \) and \( 0 \leq A \leq \frac{\pi}{4} \), we will follow these steps:

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