A way of remembering the trig formulas is to picture a right triangle in the first quadrant of the coordinate plane (positive x and y axes) with the base along the x axis and the hypotenuse starting at (0,0). The hypotenuse has length 1, label the altitude as sine and the base as cosine. Tangent is the slope of the line, equal to sine/cosine.
It always seemed to me that cosecant should be 1/cosine instead of 1/sine. I have to remember that secant and cosecant are the opposite of what I expect.
Here is how I remember the sine and cosine angle addition formulas.
sine(x+y) = sine(x) ?(y) ? cosine (x) ? (y)
sine sounds like sign, meaning that the second ? matches the + in sine(x+y). The price that you pay for having the signs match is that the two terms are heterogeneous – they mix sine and cosine, so in the end you get:
sine(x+y) = sine(x)cos(y) + cosine(x)sine(y)
For cosine(x+y) you have: cosine(x+y) = cosine(x) ?(y) ? sine(x) ?(y)
In this case the sign is opposite, meaning the second ? is -. The reward for this is having homogeneous terms. cosine is with cosine and sine is with sine, giving:
cos(x+y) = cosine(x)cosine(y) – sine(x)sine(y)