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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
⇒
=
=