# How do you prove that sec(pi/3)tan(pi/3)=2sqrt(3)?

Feb 1, 2015

You can start by breaking down your left side of the identity calculating each trigonometric function and see what happens.
Remember that pi/3=60° and this is a "special" angle which has "known" values of $\sin , \cos , \tan , \ldots$ etc.

Now:

$\sec \left(\frac{\pi}{3}\right) = \frac{1}{\cos \left(\frac{\pi}{3}\right)} = \frac{1}{\frac{1}{2}}$
$\tan \left(\frac{\pi}{3}\right) = \sin \frac{\frac{\pi}{3}}{\cos \left(\frac{\pi}{3}\right)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}$
Let's put them all together:
$\sec \left(\frac{\pi}{3}\right) \tan \left(\frac{\pi}{3}\right) = \frac{1}{\frac{1}{2}} \cdot \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} =$
manipulate your fractions to get:
$= 4 \cdot \frac{\sqrt{3}}{2} = 2 \sqrt{3}$
Which is indeed the result you needed to get to satisfy the identity.

Hope it helps