For the principal values, evaluate the following:
We can write,
tan = tan (2π –
)
tan(2π – θ )
= tan(–θ)
= –tanθ
∴ tan becomes –tan
–tan = –
⇒ 2tan = –
∴ The question converts to cosec–1(–)
Let cosec–1 = y
⇒ cosec y =
= cosec
The range of principal value of cosec–1 is –{0}
and cosec
Therefore, the principal value of cosec–1 is
.