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

⇒

=