# How do you find the exact value of sec165 using the half angle formula?

Jan 5, 2017

- (2)/(sqrt(2 + sqrt3)

#### Explanation:

Use the trig identity:
$2 {\cos}^{2} a = 1 + \cos 2 a .$ (1)
In this case a = 165, and 2a = 330.
sec 330 = 1/(cos 330). First, find cos (330).
$\cos \left(330\right) = \cos \left(- 30 + 360\right) = \cos \left(- 30\right) = \cos 30 = \frac{\sqrt{3}}{2}$.
Substitute in equation (1), we get:
$2 {\cos}^{2} \left(165\right) = 1 + \cos \left(330\right) = 1 + \frac{\sqrt{3}}{2} = \frac{2 + \sqrt{3}}{2}$
${\cos}^{2} \left(165\right) = \frac{2 + \sqrt{3}}{4}$
$\cos \left(165\right) = \pm \frac{\sqrt{2 + \sqrt{3}}}{2}$.
Since 165 is in Quadrant II, then, we take the negative value.
sec (165) = 1/(cos) = - 2/(sqrt(2 + sqrt3) =