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This means you have a positive cos and negative tan.

This means theta is in quadrant IV which means it is in the range of -pi/2 to 0. Your options are pi/4 and 7pi/4 but these don't fit in your range. However, you know you can always add or subtract 2pi and all the trig functions will be the same. pi/4 - 2pi = -7/4pi which is still outside the known range.

7pi/4 - 2pi = -pi/4 which is in the range and must be the answer.

Remember, any time they say sec(theta) = x, then there are an infinite number of values for theta that will work. You just have to find the one in the correct range.