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^{–1}is [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

⇒

=

=

3