Sec(-585°) = Sec(585°) since secant is an even function.

585° can be rewritten as 585° - 2 × 360° = -135°.

Sec(-135°) = Sec(135°) since secant is an even function.

Sec(135°) = Sec(45° + 90°) = -Sec(45°) since secant has a period of 360° and is negative in the second quadrant.

Sec(45°) = √2.

Therefore, Sec(-585°) = -√2.

To calculate sec(-585°) we use the periodicity of secant.

The period of the function secant is 360°. Then, sec(x) = sec(x + 360°).

Thus, sec(-585°) = sec(-585° + 360°) = sec(-225°).

We can further simplify,

sec(-225°) = sec(135°).

Then we can apply the following identity, sec(x) = 1/cos(x)

Therefore sec(135°) = 1 / cos(135°).

We already know that cos(135°) = -√2/2.

So, sec(135°) = 1 / -√2 / 2 = -2 / √2 = -√2

Therefore, sec(-585°)= -√2.