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First we will convert those radian angles to degrees, since my mind works better with degrees. Let's work one at a time. First, 
. If we start at the positive x-axis and measure out 315 we end up in the 4th quadrant with a reference angle of 45 with the positive x-axis. The side across from the reference angle is -1, the side adjacent to the angle is 1, and the hypotenuse is sqrt2. The cotangent of this angle, then is 1/-1 which is -1. As for the second one, converting radians to degrees gives us that 
. Sweeping out that angle has us going around the origin more than once and ending up in the first quadrant with a reference angle of 30° with the positive x-axis. The side across from the angle is 1, the side adjacent to the angle is √3, and the hypotenuse is 2. Therefore, the secant of that angle is 2/√3.
Call the weights of the two dogs 
and 
.
We know that Tripp weights 35lbs more than Rico, or:
We also know that the total weight of both dogs is 49lbs.:
Now, by substitution:
Rico weighs 7lbs.
Then:
Tripp weighs 42lbs.
To check our work:
CHECK!
We know that Tripp weights 35lbs more than Rico, or:
We also know that the total weight of both dogs is 49lbs.:
Now, by substitution:
Rico weighs 7lbs.
Then:
Tripp weighs 42lbs.
To check our work: