Question

What are all the solutions to `secx= -1/2`?

Answer 1

There are no real solutions for which `secx=-1/2`

Explanation
`secx=-1/2 => cosx=-2`

Whereas the range of validity of `cosx` is `-1<=cosx<=1`

So there is no angle `x` for which `cosx=-1/2`

Even from the graph this can be seen,
graph{secx [-10, 10, -2, 2]}