For the principal

The Principal value for tan^–1√3

Let tan^–1(√3 ) = y

⇒ tan y = √3

The range of principal value of tan^–1is {}

And tan = √3

∴ The principal value of tan^–1(√3 ) is .

Now,

Principal value for sec^–1(–2)

Let sec^–1(–2) = z

⇒ sec z = –2

= – sec = 2

= sec

= sec

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

and sec = –2

Therefore, the principal value of sec^–1(–2 ) is .

∴ tan^–1√3 –sec^–1(–2)

=

=

∴ tan^–1√3 – sec^–1(–2) = .