How am I supposed to work this out, or do I have to memorize? When simplifying a trigonometric expression, say, $\sin^2 \theta$ / $\cos^2 \theta$ - I remember that sin over cos is equal to tan. 
However, what other identities, such as the one mentioned above, do I need to know in general? 
Is their a way to quickly work them out? or do you have to memorize them?
Thanks and regards,
 A: There are several useful identities. However these basic once you remmember as soon as you understand what it actually means. 
$ \cos t =\frac{\text{adj}}{\text{hyp}}$
$ \sin t =\frac{\text{opp}}{\text{hyp}}$
$ \tan t =\frac{\sin }{\cos }$=$\frac{\text{opp}}{\text{adj}}$

We know from the phytagorean Theorem Identity that 
$\sin ^2(x)+\cos ^2(x)=1$ By dividing by $\sin ^2(x)$ or $\cos ^2(x)$ we are getting two useful alternative versions of that identity. 
Other useful identities are double angle identities, you are not expected to remember all identities. However some of them you end up using so often that they become second nature. 
A: Here is an exercise that may help you out.

The top triangle is actually 4 similar triangles being overlaid. On the bottom is a pink triangle; on top of that is an orange triangle, a beige one, and a green one. Below the big triangle I have shown the smaller triangles in isolation.
Your task is 3-fold.


*
  
*find the length of the unknown sides in the orange, beige, and green triangles (express them in terms of trigonometric functions of $t$)
  
*write out the definitions of each trigonometric function (cos, sin, tan, csc, sec, cot) using each of the 4 triangles (that's 4 sets of definitions).
  
*write out the Pythagorean Theorem for each of the 4 triangles.

