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