Help with a trig problem?

Question

Evaluate the following expression: cot [csc^-1 (13/5)]

I haven't done trig in a while and can't quite remember how to do this. Any help would be appreciated and please explain so that I can fully understand. 10 points to the best answer!

Answer 1

Let x = csc^(- 1)(13 / 5).

Then:

csc(x) = 13 / 5

and x is in quadrant 1, as csc^(- 1) has domain [- pi / 2, pi / 2].

1 + cot^2(x) = csc^2(x) = 13^2 / 5^2

cot^2(x) = 13^2 / 5^2 - 1

= 12^2 / 5^2 (5-12-13 triangle).

In quadrant 1, cot is +ve.

Therefore:

cot(x) = 12 / 5.