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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
⇒
=