The trigonometric identity secant is equal to:
[tex]\sec \alpha=\frac{1}{\cos \alpha}[/tex]
In the unit circle, the cosine is given by the x-coordinate, then, here we have the unit circle:
As we said before, the cosine of an angle is the x-coordinate, then:
[tex]\cos \frac{3\pi}{4}=\frac{-\sqrt[]{2}}{2}[/tex]
And, then the secant is:
[tex]\begin{gathered} \sec \frac{3\pi}{4}=\frac{1}{\cos \frac{3\pi}{4}} \\ \sec \frac{3\pi}{4}=\frac{1}{\frac{-\sqrt[]{2}}{2}} \\ By\text{ applying the properties of fractions:} \\ \sec \frac{3\pi}{4}=\frac{2}{-\sqrt[]{2}} \\ \sec \frac{3\pi}{4}=\frac{2}{-\sqrt[]{2}}\cdot\frac{-\sqrt[]{2}}{-\sqrt[]{2}} \\ \sec \frac{3\pi}{4}=\frac{-2\sqrt[]{2}}{2} \\ \text{Simplify 2/2=1} \\ \sec \frac{3\pi}{4}=-\sqrt[]{2} \end{gathered}[/tex]
The answer is the last option.
