"reciprocal" of anything is "1 divided by" that anything.
And "sec theta" is the reciprocal of "cos theta".
So "sec theta = - root 2" is the same thing as "cos theta = 1 / (-root 2)".
1 /root 2 is one of those well-known ones, cos theta = 1 / root 2 means that theta = 45 degrees.
But it's minus, so you have to go into the 2nd or 3rd quadrant (where x is negative) which gives theta as 135 or 225 degrees.
Now cot theta = infinite.
Cot theta is the reciprocal of tan theta, so cot theta = infinite is the same as tan theta = 1 / infinite which is zero.
(No complicating arguments about what this "means" - we're in basic stuff here.)
So we have tan theta = 0 which means theta = 0 or theta = 180 degrees.
Hope this helps.