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Prove that: tan^(-1)(1/2tan2A)+tan^(-1)(...

Prove that: `tan^(-1)(1/2tan2A)+tan^(-1)(cotA)+tan^(-1)(cot^3A) =0`

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For `0 lt A lt (pi//4), cot A gt 1`
`rArr (cot A) (cot^(3) A) gt 1`
Then, `tan^(-1) ((1)/(2) tan 2A) + tan^(-1) (cotA) + tan^(-1) (cot^(3) A)`
`=tan^(-1) ((tanA)/(1 - tan^(2)A)) + pi + tan^(-1) ((cot A + cot^(3) A)/(1 - cot^(4)A))`
`= tan^(-1) ((tan A)/(1 - tan^(2)A)) + pi + tan^(-1) ((cot A)/(1- cot^(2)A))`
`= tan^(-1) ((tanA)/(1- tan^(2)A)) + pi + tan^(-1) ((tanA)/(tan^(2) A -1)) = pi`
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