Question

​Find the principal value of the following:

`cosec^-1(-sqrt2)`

Answer 1

Let `cosec^-1(-sqrt2)=y`

Then,

`cosecy=-sqrt2`

We know that the range of the principal value branch is `[-pi/2,pi/2]-{0}`.

Thus,

`cosecy=-sqrt2=cosec(-pi/4)`

`y=-pi/4in [-pi/2,pi/2],y!=0`

Hence, the principal value of `cosec^-1(-sqrt2)   is   -pi/4.`