Do you know what the functions sin and cos are? You probably should but if you don't, here is something that may help Trigonometric functions - Wikipedia, the free encyclopedia. The stuff inside the functions is just a times x, where a is a constant, usually a real number.
Symbolic manipulator is a program like Mathematica, which lets you plug in equations without number (such as the above equation) and you can literally manipulate the expressions. In Mathematica for example, there are powerful algorithms that let you factor and simplify expressions and also transform expression using known identities. A CAS is a computer program with which you can do numerical math calculations. Nowadays a CAS and symbolic manipulator come hand in hand. Mathematica and Maple are the one I use.
In part a), you need to use trig identities for the double angles and for the sums of angles to express y as a sum of products of sine and cosine functions that involve only a single x. Once you do this for the first couple of values of a, you should see a clear pattern. Then you need to conjecture a formula.
In part c) you need to use a CAS to manipulate either the conjectured formula or simplify y.