For the principal values, evaluate the following:

Let sec^–1(–√2) = y

⇒ sec y = –√2

= – sec = √2

= sec

= sec

The range of principal value of sec^–1is [0, π]–{}

and sec = –√2.

Let,

cosec^–1–√2 ⁼ z

⇒ cosec z = –√2

⇒ –cosec z = √2

⇒ –cosec = √2

As we know cosec(–θ) = –cosecθ

∴ –cosec = cosec

The range of principal value of cosec^–1 is –{0} and

cosec = –√2

Therefore, the principal value of cosec^–1(–√2) is .

cosec^–1–√2 ⁼ y

⇒ cosec y = –√2

⇒ –cosec y = √2

⇒ –cosec = √2

As we know cosec(–θ) = –cosecθ

∴ –cosec = cosec

The range of principal value of cosec^–1 is –{0} and

cosec = –√2

Therefore, the principal value of cosec^–1(–√2) is .

From (1) and (2) we get

⇒

=

=