First, draw the chart below with SIN TAN SEC on left, 
CO-functions on right, and 1 in the middle.

        +---------------+
        |               |
        |  SIN-----COS  |
        |    \     /    |
        |     \   /     |
        |      \ /      |
        |  TAN--1--COT  |
        |   |  / \  |   |
        |   | /   \ |   |
        |   |/     \|   |
        |  SEC     CSC  |
        |               |
        +---------------+



Using this chart (I just look at it in my head) you can remember the following things:

Across the 1:
        1/SIN = CSC  or  1/CSC = SIN
        1/TAN = COT  or  1/COT = TAN
        1/SEC = COS  or  1/COS = SEC

Down any triangle:
        SIN2 + COS2 = 1
        TAN2 + 1     = SEC2
            1 + COT2 = CSC2

Up any triangle:
        SEC2-1     = TAN2  or      1-TAN2 = SEC2
        CSC2-1     = COT2  or  CSC2-COT2 = 1
            1-SIN2 = COS2  or      1-COS2 = SIN2

A function and its two nearest CLOCKWISE or COUNTERCLOCKWISE neighbors
around any edge of the square:
        (listed starting at TAN going clockwise)
                TAN = SIN/COS
                SIN = COS/COT
                COS = COT/CSC
                CSC = SEC/TAN
                SEC = TAN/SIN
        (listed starting at TAN going counter-clockwise)
                TAN = SEC/CSC
                SEC = CSC/COT
                CSC = COT/COS
                COS = SIN/TAN
                SIN = TAN/SEC

A function and its two neighbors around any edge of the square:
        (listed starting at TAN going clockwise)
                TAN = SIN*SEC
                SIN = COS*TAN
                COS = COT*SIN
                CSC = COT*SEC
                SEC = TAN*CSC


Don't forget to put in the arguments in each of the identities above!!