Find the principal values of each of the following:

Let us assume 2tan = θ

We know tan = –1

∴ 2tan = 2(–1)

⇒ 2tan = –2

∴ The question converts to sec^{–1}(–2)

Now,

Let sec^{–1}(–2) = y

⇒ sec y = –2

= – sec = 2

= sec

= sec

The range of principal value of sec^{–1}is [0, π]–{}

and sec = –2

∴ The principal value of sec^{–1}(2tan) is

1