Evaluate each of the following: `cot^(-1)(1/(sqrt(3)))-cos e c^(-1)(-2)+sec^(-1)(2/(sqrt(3)))` `cot^(-1){2"cos"(sin^(-1)((sqrt(3))/2))}+ cos e c^(-1)(-2/(sqrt(3)))+2cot^(-1)(-1)` `tan^(-1)(1/(sqrt(3)))+cot^(-1)(1/(sqrt(3)))+tan^(-1)(sin(-pi/2))`

Explanation

(i) `cot^-1 (1/sqrt3) - cosec^-1(-2) + sec^-1(2/sqrt3)`

`= pi/3 - (-pi/6)+ pi/6= pi/3 + pi/6 + pi/6 = (2pi)/3`

(ii) `cot^-1(2cos(sin^-1 (sqrt3/2)))`

`= cot^-1 { 2cos(pi/3)}`

`= cot^-1{1}= pi/4`

(iii) `cosec^-1 (-2/sqrt3) + 2cot^-1(-1)`

`= -pi/6 + 2(-pi/4) = - pi/6 + (-pi/2)`

`= -(2pi)/3`

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