Question

How do you solve `secx/cosx - 1/2secx= 0`?

Answer 1

There are no real solutions

Explanation:

`sec A = 1/cos A`

`sec x/cos x - 1/2sec x = 0`
`sec x(sec x - 1/2) = 0`
`sec x = 0` or `sec x = 1/2`

The range of `|sec x| >= 1`, so there are no real solutions

graph{sec x [-10, 10, -5, 5]}

Answer 2

Maple software produces this answer

`x=arccos(2)`

`x` is approximately `1.317i`

NOTE: This is not a real solution as the other poster dani83 states. If you are restricting yourself to the real numbers then dani83 is correct.

Explanation:

Rewrite
NOTE: `sec(x)=1/cos(x)`

`1/cosx(1/cosx)-1/2sec(x)=0`

This is just

`sec^2x-1/2secx=0`

Factor

`secx(secx-1/2)=0`

`secx` cannot be zero so

`secx-1/2=0`

`secx=1/2`

Take reciprocal

`cosx=2`

`arccos(cosx)=arccos(2)`

`x=arccos(2)`

This is not a real solution as the other poster says