Find the value of `2cos^(-1)(3/(sqrt(13)))+cot^(-1)(16/63)+1/2cos^(-1)(7/25)`

Explanation

`E = 2 cos^(-1).(3)/(sqrt13) + cot^(-1).(16)/(63) + (1)/(2) cos^(-1).(7)/(25)`
`= 2 tan^(-1).(2)/(3) + tan^(-1).(63)/(16) + (1)/(2) cos^(-1).(7)/(25)`
Now, `2 tan^(-1).(2)/(3) = tan^(-1). (2 ((2)/(3)))/(1 - (4)/(9))`
`= tan^(-1).(12)/(5)`
Let `(1)/(2) cos^(-1).(7)/(25) = tan^(-1) x`
`rArr cos^(-1).(7)/(25) = 2 tan^(-1) x`
`rArr tan^(-1).(24)/(7) = tan^(-1).(2x)/(1 - x^(2))`
`rArr (24)/(7) = (2x)/(1 - x^(2))`
`rArr 12x^(2) + 7x - 12 = 0`
`rArr (4x -3) (3x + 4) = 0`
`rArr x = 3//4`
`:. E = tan^(-1).(12)/(5) + tan^(-1).(63)/(16) + tan^(-1).(3)/(4)`
`= pi + tan^(-1).((12)/(5) + (3)/(4))/(1-((12)/(5)) ((3)/(4))) + tan^(-1).(63)/(16)`
`= pi + tan^(-1) (-(63)/(16)) + tan^(-1).(63)/(16)`
`= pi`