Show that -tan(-theta)/sec(theta) = sin(theta). For each step, explain what identity you are using.  Solve the equation cot^2(theta)-3csc(theta) = 0 for all solutions of theta, showing all steps. Use set notation and K € Z to index solutions. Leave answers in exact form, no decimals.  Solve the equation sin(theta)+3cos(theta) = -1 for theta € [0,2pi).  There are at least 3 different ways to solve the equation (sec^2(2theta)-1)^2 = 9. Find at least 2 of those ways and solve for theta € [0,2pi). Let v = 2theta in both examples. What interval is v in to get theta € [0,2pi)? One way should include Pythagorean identity, but no factoring. The other example should include factoring (another substitution may help), but not a Pythagorean identity. Show all steps in process. Compare results to make sure solutions are the same and illustrate on a unit circle. For the 2 examples used, there is a different trigonometric function in the last equation before finding actual solutions. For each one, graph the trigonometric function with variable theta for [0,2pi) and with v for the interval found above, on the same axes to illustrate the difference in the period. Explain how the graphs are related. Illustrate the solutions for both theta and v on the graphs of both trig functions. Explain the relationships between the solutions on each graph.