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

Question

3881 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-07-25T17:37:46

$- \sqrt{2} .$ $-sqrt2.$

Recall that,

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

We know that,

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

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

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