Asap given that tan theta = -1 , what is the value of sec theta, for 3pi/2 < theta , 2 pi

The tan is negative in the 2nd and 4th quadrants, knowing this, the angle that equals the value that tan = 1 in the first quadrant is the tan 45° = 1, but the question restrict for 3pi/2 < theta < 2pi, so it's the same as 270° < theta < 360° (4° Quadrant), then moving the tan 45° = to this quadrant the value will be negative, tan in 4°Q = tan(360 - 45=315°) = -1.

sec theta = 1/cos theta

cos 45° =


cos values in 4°Q will be positives, so it doesn't change the signal.

sec 315° = 1/cos 315°
sec 315° = 1/square root (2)/2
sec 315 = 1/1 x 2/sqroot(2)