Evaluate each of the following:

Now first of the principal value of

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 …(1)

Now, the value of cot^{–1}(–1)

Let cot^{–1}(–1) = y

⇒ cot y = –1

= – cot = 1

= cot

= cot

The range of principal value of cot^{–1}is (0, π)

and cot = –1

Therefore, the principal value of cot^{–1}(–1) is …(2)

From (1) and (2) we can write the given equation as

=

=

=

3