I need to solve this equation and graph it using critical points, 2nd derivative test, etc. I'm not well versed on trig at all so I have no idea where to start or how I'll even complete this problem.

Given:

y = \cos (x) -\frac 1 2 \cos (2x)

you can find the roots of this equation by substituting the trig identity \cos (2x) = 2 \cos^2(x) - 1, and then you can determine the values for cos(x) that satisfies 0 = \cos x - \frac 1 2 (2 \cos^2x - 1). Then from that you can determine the values for x. Unfortunately the answer(s) turn out to not be "pretty."

An easier task is to find the local minima and maxima. Determine the equation for dy/dx and set it to equal 0, and solve for x. You should get an equation that includes a factor of sin(2x). You can solve this by substituting the identity sin2x = 2 sinx cosx.