Answer: [tex]-\sqrt{2}[/tex]
Step-by-step explanation:
The trigonometric function secant is the reciprocal of cosine:
[tex]sec(\alpha)=\frac{1}{cos(\alpha)}[/tex]
For [tex]-135\°[/tex]:
[tex]sec(-135\°)=\frac{1}{cos(-135\°)}[/tex] (1)
On the other hand, it is known [tex]cos(-\alpha)=cos(\alpha)[/tex], hence:
[tex]cos(-135\°)=cos(135\°)[/tex] (2)
In addition, it is known [tex]cos(135\°)=-cos(45\°)=-\frac{\sqrt{2}}{2}[/tex] (3)
Substituting this on (1):
[tex]sec(-135\°)=\frac{1}{-\frac{\sqrt{2}}{2}}[/tex] (4)
[tex]sec(-135\°)=-\frac{2}{\sqrt{2}}[/tex] (5)
[tex]sec(-135\°)=-\frac{2}{\sqrt{2}}(\frac{\sqrt{2}}{\sqrt{2}})[/tex] (6)
Finally:
[tex]sec(-135\°)=-\sqrt{2}[/tex]
