Suppose that sec(t)= -5/4 and that t is in the 3rd quadrant. Find the exact value of tan(t).

Using the identity that

1 + tan^2 = sec^2 ..so....ubtract 1 from both sides

tan^2 = sec^2 - 1

tan^2 = (-5/4)^2 - 1

tan^2 = 25/16 - 1

tan^2 = 9/16 since the tangent is positive in the 3rd quadrant....take the positive root

tan = √ [9/16] = 3/4