Question

How do you evaluate `csc(arctan(-1))` without a calculator?

Answer 1

`-sqrt2.`

Explanation:

Recall that,

`arc tanx=theta, x in RR iff tan theta=x, theta in (-pi/2,pi/2).`

We know that,

`tan(-pi/4)=-1, &, -pi/4" lies in "(-pi/2,pi/2). :. arc tan(-1)=-pi/4.`

`:." The Reqd. Value="csc(arc tan (-1))=csc(-pi/4)=-sqrt2.`