Prove that: `cot^(-1)((ab+1)/(a-b))+cot^(-1)((bc+1)/(b-c))+cot^(-1)((ca+1)/(c-a))=0`.

Prove that: `cot^(-1)((ab+1)/(a-b))+cot^(-1)((bc+1)/(b-c))+cot^(-1)((c

1.	Prove that: `cot^(-1)((ab+1)/(a-b))+cot^(-1)((bc+1)/(b-c))+cot^(-1)((ca+1)/(c-a))=0`.
Answer» We have, LHS `=tan^(-1)(a-b)/(1+ab)+tan^(-1)(b-c)/(1+bc)+tan^(-1)(c-a)/(1+ca)` `=(tan^(-1)a-tan^(-1)b)+(tan^(-1)b-tan^(-1)c)+(tan^(-1)c-tan^(-1)a)` =0=RHS `therefore` LHS=RHS.

Range of `"cosec"^(-1)x` isA. `(-pi/2,pi/2)`B. `[-pi/2,pi/2]`C. `[-pi/2,pi/2]-{0}`D. none of these
The value of `"cosec"^(-1)("cosec"(4pi)/3)` isA. `(3pi)/4`B. `pi/4`C. `-pi/4`D. none of these
Find the principalvalue of `sec^(-1)((-2)/(sqrt(3)))`A. `pi/6`B. `-pi/6`C. `(5pi)/6`D. `(7pi)/6`
The principal value of `"cosec"^(-1)(-sqrt(2))` isA. `-pi/4`B. `(3pi)/4`C. `(5pi)/4`D. none of these
Find the principal value of each of the following: i) `cos^(-1)(-1/sqrt(2))`, ii) `cos^(-1)(-1/2)`, iii) `cot^(-1)(-1/sqrt(3))`
Find the principal value of each of the following: i) `cot^(-1)(-sqrt(3))` ii) `sec^(-1)(-sqrt(2))` iii) `"cosec"^(-1)(-1)`.
The principal value of `cot^(-1)(-1)` isA. `-pi/4`B. `pi/4`C. `(5pi)/4`D. `(3pi)/4`
Find the principal valuesof each of the following:`cot^(-1)(-sqrt(3))`(ii) `cot^(-1)(sqrt(3))`A. `-pi/6`B. `pi/6`C. `(7pi)/6`D. `(5pi)/6`
If `tan^(-1)x=pi/4-tan^(-1)(1/3)` then `x` isA. `1/2`B. `1/4`C. `1/6`D. none of these
The value of `tan^(-1)(tan(8*pi/6))` isA. `(7pi)/6`B. `(5pi)/3`C. `pi/3`D. none of these