निम्नलिखित समीकरणों को सरल कीजिए : `2 tan^(-1)(cosx)=tan^(-1)(2 "cosec"x)`

निम्नलिखित समीकरणों को सर�

1.	निम्नलिखित समीकरणों को सरल कीजिए : `2 tan^(-1)(cosx)=tan^(-1)(2 "cosec"x)`
Answer» We have, `2tan^(-1)(cosx)=tan^(-1)(2"cosec"x)` `rArr tan^(-1)(2cosx)/(1-cos^(2)x)=tan^(-1)(2"cosec"x)` `rArr tan[tan^(-1)(2cosx)/(sin^(2)x)]=2"cosec"x`. `rArr (2cosx)/(sin^(2)x)=2"cosec"x rArr cosx=sinx` `rArr tanx=1 rArr x=pi/4`.

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