Prove that:
`tan^(-1)(1/2tan2A)+tan^(-1)(cota)+tan^(-1)(cot^3A)=``{0,ifpi/4 lt a lt pi/2pi`,`if0 lt A lt pi/4}`

Explanation

`tan ^{-1}(frac{1}{2} tan 2 {~A})+tan ^{-1}(frac{cot {A}+cot 3 {~A}}{1-cot 4 {~A}})`
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tan ^{-1}(frac{2 tan {A}}{2(1-tan ^{2} {x})})+tan ^{1}(frac{cot {A}}{1-cot ^{2}})`
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Rightarrow tan ^{-1}(frac{tan {A}}{1-tan ^{2} {~A}})+tan ^{-1}(frac{frac{1}{tan {A}}}{tan ^{2} {~A}-1})`
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.Rightarrow tan ^{-1}(frac{tan {A}}{1-tan ^{2} {~A}})+tan ^{-1})(frac{tan {A}}{tan ^{2} {~A}-1})`
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Rightarrow if frac{pi}{4} lt {A} lt frac{pi}{2}`
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Rightarrow tan -1(frac{tan {A}}{1-tan ^{2} {~A}})-tan ^{-1}(frac{tan {A}}{1-tan ^{2} {~A}})=0`
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if 0 lt {A} lt frac{pi}{2}`
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Rightarrow tan -1(frac{tan {A}}{1-tan ^{2} {~A}})+cot ^{-1}(frac{tan {A}}{1-tan ^{2} {~A}})
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