a) show that (cosx)^2=8(sinx)^2-6sinx can be written as (3sinx-1)^2=2 b)Solve (cosx)^2=8(sinx)^2-6sinx

Start with a). Looking at this equation, the trig identity screaming out is cosx^2+sinx^2=1--> cosx^2=1-sinx^2. Substituting this into the LHS of the equation, and with a bit of algebraic rearrangement, we get 9sinx^2-6sinx=1. This is similar to what we need, but not quite;we've hit a wall. What i would do here is then start from what we are given, and expand (3sinx-1)^2=2. This gives us 9sinx^2-6sinx+1=2. This is close to what we got from starting with the equation! Notice that if we add 1 to both sides, we get the expanded version of the result, and part a) is complete. b) Using (3sinx-1)^2=2 instead of the nasty long trig equation, we can square root both sides. Remember that when you square root something, you need to kind in mind that both the positive and negative values are valid, so we get 3sinx-1=+-root2. with a bit of manipulation, we can obtain sinx=(1+root2)/3 and sinx=(1-root2)/3. Now i recommend drawing the sin curve to help visualise what is going on. You need 2 solutions for each of the sinx=.... Using your calculator, if you arcsin each of the answers, you get 2 solutions, but we want 2 more. The curves above and below the x axis on the sin curve are both symmetrical, so the answers have to be the same distance from the ends of the curves as each other. I will demonstrate what i mean using the whiteboard. The first answer, from sinx=(1+root2)/3 is 53.58. This it 53.58 from 0, so it needs to be 53.58 from 180 (the other end of the curve. So 180-53.58=126.42. For the second answer, the arcsin gave us 187.94, which is 7.94 from one end of the curve. To find the other answer, we need to take 7.94 from 360, 352.06. So our 4 solutions are 53.58,126.42,187.94,352.06.