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 .