For the principal values, evaluate the following:

tan^{–1}√3 – sec^{–1}(–2)

The Principal value for tan^{–1}√3

Let tan^{–1}(√3 ) = y

⇒ tan y = √3

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

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^{–1}is [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) = .

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