THere are 2 pneumonic devices that can help you.
"ALL STUDENTS TAKE CALCULUS" or
"ADD SUGAR TO COFFEE"
A: all trig functions are positive in quadrant 1
S: only the SINE function is positive in quadrant 2
T: only the TANGENT function is positive in quadrant 3
C: only the COSINE function is positive in quadrant 4
So the tangent function is negative in quadrants 2 and 4.
Since the secant is given to be negative, it's reciprocal,
the cosine must also be negative. This happens in quadrants
2 and 3
Hence, the common quadrant where tangent and cosine are negative is quadrant 2.
The BASE angle where tangent is 1 is at flavors of 45 degrees.
In the QUadrant 2, that is 180-45 = 135.
tan(135) = -1 and sec(135) = -sqrt(2)