# Trig Identities #3 Periodicity

You can hopefully see from the diagram above, that if we add $2\pi$ to our
angle $\theta$, we would come ‘full circle’ (ba-dum tish) and be back where we started from (think of it as adding 360 degrees if it makes you happier).

Hence:

$(\cos(\theta \pm 2\pi), \sin(\theta \pm 2\pi)) = (\cos(\theta), sin(\theta))$

$\cos(\theta) = \cos(\theta \pm 2\pi)$

$\sin(\theta) = \sin(\theta \pm 2\pi)$

And since csc and sec are basically just utilisations of sin and cos, then they have the same property:

$\csc(\theta) = \csc(\theta \pm 2\pi)$

$\sec(\theta) = \sec(\theta \pm 2\pi)$

Now  $\tan \theta$ has a periodic cycle half the size of $\sin \theta$, which you can perhaps visualise easily by studying their respective plots side by side:

and so :

$\tan(\theta) = \tan(\theta \pm \pi)$

$\cot(\theta) = \cot(\theta \pm \pi)$