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

Explanation

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`