Question

find principal value of sec^-1(-2)

Answer 1

Let the answer be x.

Since sec−1(2)=x, sec(x)=2.

As secant is the reciprocal of the cosine function, i.e. sec(x)=1cos(x), the following must be true as well.

1cos(x)=2

Or cos(x)=12.

The range of sec−1(x) is [0,π2)∪(π2,π].
The range of cos−1(x) is [0,π].

Since the ranges are similar, we can find x using

x=cos−1(12)=π3

Answer 2

sec^-1(-2)

sec-1{(sec(pie/3)} here, sec(-theta)=sec theta

pie/3