#1. We are told that sin(x)=5/13, and x is in Quadrant I. The first thing to do is find cos(x). You could use the Pythagorean Identity, or set up a triangle that says sin(x)=5/13, or several other options, but find cos(x) first. And since x is in Quadrant I, we know what the sign for cos(x) is, too. Now we can use the double angle formulas for sine & cosine to find sin(2x) and cos(2x), and then we can take their ratio to get tan(2x).
#2-8. Same suggestions as for #1, but in each case you will need to first find sin(x) and/or cos(x), probably using a triangle. Don't forget to adjust the + or 1 sign, based on the quadrant.
#15-22. Since you are going to use the half-angle formulas, you will first need to calculate A, which is double the angle of interest. For example, to find cos(22.5o), use A = 2*22.5o = 45o. Now plug that angle into the appropriate half-angle formula to find the value of the trig function. Lastly, adjust the + or - sign to match the quadrant of the angle in question.
#23-28. Look at the list of formulas to see which one matches up best with the expression of interest. Then identify (based on that formula) what the expression must equal. For example, for 2*sin(18o)*cos(18o), that matches up with the double angle formula for sine, with angle A = 18o. From the formula, that should equal sin(2A), which equals ???
#29-34. Since you will be using the half-angle formulas here, the first thing you will need to find is cos(x) (like the hint for #1 above). From that we can find sin(x/2) and cos(x/2), and then we can take the ratio of those two to get tan(x/2).
#53-70. In each case, we need to verify the identity. That is, we need to manipulate one side of the equation to make it look like the other side. (Or we could manipulate both sides to meet somewhere in the middle.) Remember that one of our strategies was to get a common argument. In most (all?) of these problems you will need to use one or more of the formulas on our list to get a common argument. For example, on #53, one side uses 5x, and the other side uses 10x. So we could use a double angle formula on cos(10x) to change it to involve 5x instead. For another example, on #63, since one side uses x and the other uses 2x, we could use double angle formula on cos(2x) to get the arguments to match. You might also notice that the left-hand side of the identity factors (difference of squares).
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