Find the principal solutions of the following equations :
(1) `cosec x =2`
(2) `cotx=sqrt(3)`.

Explanation

The given equation is `cosec x=2` which is same as `sin x=(1)/(2)`.
we know that,
`sin.(pi)/(6)=(1)/(2)and sin.(pi-x)=sinx`
`therefore sin.(pi)/(6)=sin(pi-(pi)/(6))=sin,(5pi)/(6)`
`therefore sin.(pi)/(6)=sin.(5pi)/(6)=(1)/(2)`, where
`0lt (pi)/(6)lt 2pi and 0 lt (5pi)/(6)lt 2pi`
`therefore sinx=(1)/(2)` gives,
`sin x =sin.(pi)/(6)=sin.(5pi)/(6)`
`therefore x=(pi)/(6)and x=(5pi)/(6)`
Hence , the required principal solutions are
`x=(pi)/(6) and x=(5pi)/(6)`.
(2) .The given equation is `cotx=sqrt(3)` which is same as `tanx=(1)/(sqrt(3))`
We know that ,
`tan (pi)/(6)=(1)/(sqrt(3)) and tan (pi+x)=tan x`
`therefore tan.(pi)/(6)=tan(pi+(pi)/(6))=tan.(7pi)/(6)`
`therefore tan.(pi)/(6)=tan.(7pi)/(6)=(1)/(sqrt(3))`, where
`0 lt (pi)/(6)lt2pi and 0 lt (7pi)/(6)lt 2pi`
`therefore cot x=sqrt(3)`, i.e., `tanx= (1)/(sqrt(3))` gives
`tanx=tan.(pi)/(6)=tan.(7pi)/(6)`
`therefore x=(pi)/(6)and x=(7pi)/(6)`
Hence , the rquried principal solutions are
`x=(pi)/(6)and x =(7pi)/(6)`.