For the principal values, evaluate the following:

Let,

Sin^{–1} ^{=} y

⇒ sin y =

⇒ –sin y =

⇒ –sin

As we know sin(–θ) = –sinθ

∴ –sin = sin

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

Therefore, the principal value of Sin^{–1} is ….(1)

Let,

cosec^{–1} ^{=} z

⇒ cosec z =

⇒ –cosec z =

⇒ –cosec

As we know cosec(–θ) = –cosecθ

∴ –cosec = cosec

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

cosec

Therefore, the principal value of cosec^{–1} is ….(2)

From (1) and (2) we get

⇒

=

3